34:
58:
249:
6414:
6169:
4963:
5453:
2763:
7320:
2138:
7233:
splits in factors that are linear or quadratic factors. Only the linear factors correspond to infinite (real) branches of the curve, but if a linear factor has multiplicity greater than one, the curve may have several asymptotes or parabolic branches. It may also occur that such a multiple linear
311:
The idea that a curve may come arbitrarily close to a line without actually becoming the same may seem to counter everyday experience. The representations of a line and a curve as marks on a piece of paper or as pixels on a computer screen have a positive width. So if they were to be extended far
6159:
are real numbers. All three types of asymptotes can be present at the same time in specific examples. Unlike asymptotes for curves that are graphs of functions, a general curve may have more than two non-vertical asymptotes, and may cross its vertical asymptotes more than once.
1485:
3958:
1629:
4255:
of the numerator and degree of the denominator determine whether or not there are any horizontal or oblique asymptotes. The cases are tabulated below, where deg(numerator) is the degree of the numerator, and deg(denominator) is the degree of the denominator.
1163:
3803:
4112:
2673:
4952:
4216:
3040:
7753:
7643:
312:
enough they would seem to merge, at least as far as the eye could discern. But these are physical representations of the corresponding mathematical entities; the line and the curve are idealized concepts whose width is 0 (see
5994:
4764:
The vertical asymptotes occur only when the denominator is zero (If both the numerator and denominator are zero, the multiplicities of the zero are compared). For example, the following function has vertical asymptotes at
3600:
3452:
2484:
2417:
6639:
1748:
6854:
5596:
7210:
4633:
7533:
5234:
1281:
7405:
5869:
6401:. An asymptote serves as a guide line to show the behavior of the curve towards infinity. In order to get better approximations of the curve, curvilinear asymptotes have also been used although the term
7872:
1837:
1031:
963:
3335:
1349:
207:
Vertical asymptotes are vertical lines near which the function grows without bound. An oblique asymptote has a slope that is non-zero but finite, such that the graph of the function approaches it as
3810:
1496:
5262:
If the degree of the numerator is more than 1 larger than the degree of the denominator, and the denominator does not divide the numerator, there will be a nonzero remainder that goes to zero as
4486:
7029:
2288:
2234:
775:
710:
6361:
4750:
236:, and determining the asymptotes of a function is an important step in sketching its graph. The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of
3279:
3197:
402:
5131:
When the numerator of a rational function has degree exactly one greater than the denominator, the function has an oblique (slant) asymptote. The asymptote is the polynomial term after
4363:
1194:(0) = 5. The graph of this function does intersect the vertical asymptote once, at (0, 5). It is impossible for the graph of a function to intersect a vertical asymptote (or
1069:
3674:
7453:
5044:
2814:
2744:
1882:
293:
7301:
359:
7083:
6085:
866:
818:
6947:
6902:
4520:
3507:
depending on the case being studied. It is good practice to treat the two cases separately. If this limit doesn't exist then there is no oblique asymptote in that direction.
8016:
4009:
1952:
489:
5703:
Although the definition here uses a parameterization of the curve, the notion of asymptote does not depend on the parameterization. In fact, if the equation of the line is
5126:
2564:
5135:
the numerator and denominator. This phenomenon occurs because when dividing the fraction, there will be a linear term, and a remainder. For example, consider the function
2101:
7962:
5745:
1308:
3505:
3482:
2067:
2044:
2530:
Functions may lack horizontal asymptotes on either or both sides, or may have one horizontal asymptote that is the same in both directions. For example, the function
2001:
4669:
1208:
If a function has a vertical asymptote, then it isn't necessarily true that the derivative of the function has a vertical asymptote at the same place. An example is
7899:
1341:
5070:
4783:
4390:
4305:
2127:
1911:
606:-axis) near which the function grows without bound. Oblique asymptotes are diagonal lines such that the difference between the curve and the line approaches 0 as
4121:
7923:
6157:
2021:
1972:
1653:
533:
513:
462:
442:
422:
2884:
7657:
3345:
The asymptotes of many elementary functions can be found without the explicit use of limits (although the derivations of such methods typically use limits).
7547:
6449:
in this manner. Asymptotes are often considered only for real curves, although they also make sense when defined in this way for curves over an arbitrary
1205:
A common example of a vertical asymptote is the case of a rational function at a point x such that the denominator is zero and the numerator is non-zero.
3609:
should be the same value used before. If this limit fails to exist then there is no oblique asymptote in that direction, even should the limit defining
8285:
The
British enciclopaedia, or dictionary of arts and sciences; comprising an accurate and popular view of the present improved state of human knowledge
5884:
3524:
3383:
7234:
factor corresponds to two complex conjugate branches, and does not corresponds to any infinite branch of the real curve. For example, the curve
2422:
2355:
404:
where x is a number other than 0. For example, the graph contains the points (1, 1), (2, 0.5), (5, 0.2), (10, 0.1), ... As the values of
6500:
5621:. From the definition, only open curves that have some infinite branch can have an asymptote. No closed curve can have an asymptote.
515:
become smaller and smaller, say .01, .001, .0001, ..., making them infinitesimal relative to the scale shown, the corresponding values of
535:, 100, 1,000, 10,000 ..., become larger and larger. So the curve extends farther and farther upward as it comes closer and closer to the
5700:-axis is also an asymptote. A similar argument shows that the lower left branch of the curve also has the same two lines as asymptotes.
5512:
8484:
4541:
1661:
7464:
5141:
1214:
7336:
424:
become larger and larger, say 100, 1,000, 10,000 ..., putting them far to the right of the illustration, the corresponding values of
5777:
594:
asymptotes depending on their orientation. Horizontal asymptotes are horizontal lines that the graph of the function approaches as
7764:
1760:
1480:{\displaystyle \lim _{x\to 0^{+}}f(x)=\lim _{x\to 0^{+}}\left({\tfrac {1}{x}}+\sin \left({\tfrac {1}{x}}\right)\right)=+\infty ,}
969:
901:
3953:{\displaystyle n=\lim _{x\rightarrow +\infty }(f(x)-mx)=\lim _{x\rightarrow +\infty }\left({\frac {2x^{2}+3x+1}{x}}-2x\right)=3}
3285:
1624:{\displaystyle \lim _{x\to 0^{-}}f(x)=\lim _{x\to 0^{-}}\left({\tfrac {1}{x}}+\sin \left({\tfrac {1}{x}}\right)\right)=-\infty }
6737:
8413:
8192:
7096:
6472:, there are a pair of lines that do not intersect the conic at any complex point: these are the two asymptotes of the conic.
8026:, because the distance to the cone of a point of the surface tends to zero when the point on the surface tends to infinity.
4407:
2239:
2185:
716:
651:
6293:
8380:
8354:
8268:
4683:
4533:
7085:, there is no asymptote, but the curve has a branch that looks like a branch of parabola. Such a branch is called a
3203:
3138:
364:
7224:
splits into linear factors, each of which defines an asymptote (or several for multiple factors). Over the reals,
1158:{\displaystyle f(x)={\begin{cases}{\frac {1}{x}}&{\text{if }}x>0,\\5&{\text{if }}x\leq 0.\end{cases}}}
3798:{\displaystyle m=\lim _{x\rightarrow +\infty }f(x)/x=\lim _{x\rightarrow +\infty }{\frac {2x^{2}+3x+1}{x^{2}}}=2}
316:). Therefore, the understanding of the idea of an asymptote requires an effort of reason rather than experience.
7758:
The distance between the hyperboloid and cone approaches 0 as the distance from the origin approaches infinity.
8507:
6952:
6109:
An asymptote can be either vertical or non-vertical (oblique or horizontal). In the first case its equation is
4311:
162:, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.
8064:
7416:
4969:
2769:
2698:
124:
1842:
4248:
has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many vertical asymptotes.
1202:
at each point where it is defined, it is impossible that its graph does intersect any vertical asymptote.
254:
8059:
7248:
322:
7034:
6040:
831:
783:
8512:
8347:
4107:{\displaystyle m=\lim _{x\rightarrow +\infty }f(x)/x=\lim _{x\rightarrow +\infty }{\frac {\ln x}{x}}=0}
20:
8216:
vol. 2, The
Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London p. 541
4492:
2668:{\displaystyle \lim _{x\to -\infty }{\frac {1}{x^{2}+1}}=\lim _{x\to +\infty }{\frac {1}{x^{2}+1}}=0.}
115:
is a line such that the distance between the curve and the line approaches zero as one or both of the
5377:
5266:
increases, but the quotient will not be linear, and the function does not have an oblique asymptote.
7967:
7212:
the curve has a singular point at infinity which may have several asymptotes or parabolic branches.
1093:
467:
6438:
5132:
5075:
444:, .01, .001, .0001, ..., become infinitesimal relative to the scale shown. But no matter how large
225:) if the distance between the two curves tends to zero as they tend to infinity, although the term
8117:
8492:
6907:
6862:
8112:
8054:
7928:
7902:
6654:
5706:
4252:
1916:
1286:
182:
8208:
6465:
3487:
3464:
2049:
2026:
296:
8481:
8086:
7323:
Hyperbolas, obtained cutting the same right circular cone with a plane and their asymptotes.
4947:{\displaystyle f(x)={\frac {x^{2}-5x+6}{x^{3}-3x^{2}+2x}}={\frac {(x-2)(x-3)}{x(x-1)(x-2)}}}
4639:
8423:
8390:
8235:
8184:
7877:
4211:{\displaystyle n=\lim _{x\rightarrow +\infty }(f(x)-mx)=\lim _{x\rightarrow +\infty }\ln x}
2072:
6446:
1317:
199:, horizontal asymptotes are horizontal lines that the graph of the function approaches as
8:
8317:
8250:
6450:
6422:
6003:
5049:
4369:
4284:
2106:
1890:
1199:
579:
548:
237:
178:
155:
128:
8212:
3035:{\displaystyle \lim _{x\to +\infty }\left=0\,{\mbox{ or }}\lim _{x\to -\infty }\left=0.}
1981:
8372:
8230:, Translated from the first Russian ed. by L. F. Boron, Groningen: P. Noordhoff N. V.,
8130:
7908:
7748:{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=0.}
6142:
6010:(a function of one real variable and returning real values). The graph of the function
2006:
1957:
1638:
1195:
518:
498:
447:
427:
407:
7638:{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1}
8409:
8376:
8350:
8301:
8264:
8188:
6442:
4245:
2751:
2693:
547:-axis are asymptotes of the curve. These ideas are part of the basis of concept of a
136:
66:
8256:
8122:
6402:
5479:
76:
8103:
Nunemacher, Jeffrey (1999), "Asymptotes, Cubic Curves, and the
Projective Plane",
361:
shown in this section. The coordinates of the points on the curve are of the form
8488:
8451:
8436:
8419:
8386:
8231:
6430:
6398:
5456:(sec(t), cosec(t)), or x + y = (xy), with 2 horizontal and 2 vertical asymptotes.
1975:
313:
151:
5989:{\displaystyle {\frac {|ax(\gamma (t))+by(\gamma (t))+c|}{\sqrt {a^{2}+b^{2}}}}}
8172:
8035:
3595:{\displaystyle n\;{\stackrel {\text{def}}{=}}\,\lim _{x\rightarrow a}(f(x)-mx)}
2747:
159:
143:
8179:
Calculus, Vol. 1: One-Variable
Calculus with an Introduction to Linear Algebra
7093:, even when it does not have any parabola that is a curvilinear asymptote. If
8501:
8260:
8177:
6007:
3447:{\displaystyle m\;{\stackrel {\text{def}}{=}}\,\lim _{x\rightarrow a}f(x)/x}
6434:
27:
7319:
6468:, as the intersection at infinity is of multiplicity at least two. For a
5684: → 0 from the right, and the distance between the curve and the
3348:
2678:
Other common functions that have one or two horizontal asymptotes include
2479:{\displaystyle \lim _{x\rightarrow +\infty }\arctan(x)={\frac {\pi }{2}}.}
2412:{\displaystyle \lim _{x\rightarrow -\infty }\arctan(x)=-{\frac {\pi }{2}}}
7539:
6418:
598:
tends to +∞ or −∞. As the name indicates they are parallel to the
57:
33:
8475:
8471:
8134:
6634:{\displaystyle P(x,y)=P_{n}(x,y)+P_{n-1}(x,y)+\cdots +P_{1}(x,y)+P_{0}}
6244:
be another (unparameterized) curve. Suppose, as before, that the curve
2346:
2142:
248:
6413:
6168:
4962:
4260:
The cases of horizontal and oblique asymptotes for rational functions
7328:
5452:
2689:
8126:
891:–1), the numerator approaches 1 and the denominator approaches 0 as
602:-axis. Vertical asymptotes are vertical lines (perpendicular to the
6386:
6090:
This parameterization is to be considered over the open intervals (
560:
551:
in mathematics, and this connection is explained more fully below.
5591:{\displaystyle \lim _{t\rightarrow b}(x^{2}(t)+y^{2}(t))=\infty .}
2762:
2152:
are horizontal lines that the graph of the function approaches as
7761:
More generally, consider a surface that has an implicit equation
4628:{\displaystyle f(x)={\frac {2x^{2}+3x+5}{x}}=2x+3+{\frac {5}{x}}}
1743:{\displaystyle f'(x)={\frac {-(\cos({\tfrac {1}{x}})+1)}{x^{2}}}}
132:
131:
and related contexts, an asymptote of a curve is a line which is
7528:{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=0.}
5229:{\displaystyle f(x)={\frac {x^{2}+x+1}{x+1}}=x+{\frac {1}{x+1}}}
3340:
1276:{\displaystyle f(x)={\tfrac {1}{x}}+\sin({\tfrac {1}{x}})\quad }
154:+ σύν "together" + πτωτ-ός "fallen". The term was introduced by
7400:{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}
5999:
which tends to zero simultaneously as the previous expression.
6661:. Vanishing of the linear factors of the highest degree term
6475:
A plane algebraic curve is defined by an equation of the form
5864:{\displaystyle {\frac {|ax(t)+by(t)+c|}{\sqrt {a^{2}+b^{2}}}}}
2137:
1063:
does not affect the asymptote. For example, for the function
45: = 0), and oblique asymptote (purple line, given by
6469:
6284:, when there is no risk of confusion with linear asymptotes.
112:
5878:) is a change of parameterization then the distance becomes
5072:. Green: difference between the graph and its asymptote for
8482:
Hyperboloid and
Asymptotic Cone, string surface model, 1872
8019:
7867:{\displaystyle P_{d}(x,y,z)+P_{d-2}(x,y,z)+\cdots P_{0}=0,}
4957:
1832:{\displaystyle x_{n}={\frac {(-1)^{n}}{(2n+1)\pi }},\quad }
1151:
1026:{\displaystyle \lim _{x\to 1^{-}}{\frac {x}{x-1}}=-\infty }
958:{\displaystyle \lim _{x\to 1^{+}}{\frac {x}{x-1}}=+\infty }
232:
Asymptotes convey information about the behavior of curves
100:
16:
Limit of the tangent line at a point that tends to infinity
3330:{\displaystyle =\lim _{x\to \pm \infty }{\frac {1}{x}}=0.}
8458:
This has a more general treatment of asymptotic surfaces.
559:
The asymptotes most commonly encountered in the study of
88:
82:
6849:{\displaystyle Q'_{x}(b,a)x+Q'_{y}(b,a)y+P_{n-1}(b,a)=0}
5506:)). Suppose that the curve tends to infinity, that is:
61:
A curve intersecting an asymptote infinitely many times.
7205:{\displaystyle Q'_{x}(b,a)=Q'_{y}(b,a)=P_{n-1}(b,a)=0,}
5286:), then the translations of it also have an asymptote.
3349:
General computation of oblique asymptotes for functions
6022:) is the set of points of the plane with coordinates (
5660: → ∞ and the distance from the curve to the
2958:
2795:
1704:
1592:
1567:
1445:
1420:
1258:
1234:
274:
7970:
7931:
7911:
7880:
7767:
7660:
7550:
7467:
7419:
7339:
7307:
with multiplicity 4, leading to the unique asymptote
7303:, but its highest order term gives the linear factor
7251:
7099:
7037:
6955:
6910:
6865:
6740:
6503:
6296:
6145:
6043:
5887:
5780:
5709:
5515:
5269:
5144:
5078:
5052:
4972:
4786:
4686:
4642:
4544:
4495:
4410:
4372:
4314:
4287:
4124:
4012:
3813:
3677:
3527:
3490:
3467:
3386:
3288:
3206:
3141:
2887:
2772:
2701:
2567:
2425:
2358:
2242:
2188:
2109:
2075:
2052:
2029:
2009:
1984:
1960:
1919:
1893:
1845:
1763:
1664:
1641:
1499:
1352:
1320:
1289:
1217:
1198:) in more than one point. Moreover, if a function is
1072:
972:
904:
834:
786:
719:
654:
645:
if at least one of the following statements is true:
521:
501:
470:
450:
430:
410:
367:
325:
257:
229:
by itself is usually reserved for linear asymptotes.
85:
26:"Asymptotic" redirects here. Not to be confused with
8314:
The elementary differential geometry of plane curves
8252:
The elementary differential geometry of plane curves
4481:{\displaystyle f(x)={\frac {2x^{2}+7}{3x^{2}+x+12}}}
4239:
491:
is never 0, so the curve never actually touches the
103:
97:
94:
8091:
An elementary treatise on the differential calculus
2283:{\displaystyle \lim _{x\rightarrow +\infty }f(x)=c}
2229:{\displaystyle \lim _{x\rightarrow -\infty }f(x)=c}
770:{\displaystyle \lim _{x\to a^{+}}f(x)=\pm \infty ,}
705:{\displaystyle \lim _{x\to a^{-}}f(x)=\pm \infty ,}
91:
79:
8176:
8010:
7956:
7917:
7893:
7866:
7747:
7637:
7527:
7447:
7399:
7295:
7204:
7077:
7023:
6941:
6896:
6848:
6633:
6355:
6151:
6079:
5988:
5863:
5739:
5590:
5228:
5120:
5064:
5038:
4946:
4744:
4663:
4627:
4514:
4480:
4384:
4357:
4299:
4210:
4106:
3952:
3797:
3594:
3499:
3476:
3446:
3329:
3273:
3191:
3034:
2808:
2738:
2667:
2519:is a horizontal asymptote for the arctangent when
2500:is a horizontal asymptote for the arctangent when
2478:
2411:
2282:
2228:
2121:
2095:
2061:
2038:
2015:
1995:
1966:
1946:
1913:both from the left and from the right, the values
1905:
1876:
1831:
1742:
1647:
1623:
1479:
1335:
1302:
1275:
1157:
1025:
957:
860:
812:
769:
704:
527:
507:
483:
456:
436:
416:
396:
353:
287:
8022:which is centered at the origin. It is called an
7458:The equation for the union of these two lines is
6392:
5624:For example, the upper right branch of the curve
8499:
6356:{\displaystyle y={\frac {x^{3}+2x^{2}+3x+4}{x}}}
5517:
4181:
4132:
4062:
4020:
3870:
3821:
3727:
3685:
3550:
3409:
3293:
3211:
3143:
2965:
2889:
2616:
2569:
2427:
2360:
2244:
2190:
1539:
1501:
1392:
1354:
974:
906:
836:
788:
721:
656:
5444:=0+2=2, and no vertical or oblique asymptotes.
5376:If a known function has an asymptote, then the
4745:{\displaystyle f(x)={\frac {2x^{4}}{3x^{2}+1}}}
2836:When a linear asymptote is not parallel to the
6425:(solid) with a single real asymptote (dashed).
5274:If a known function has an asymptote (such as
3274:{\displaystyle =\lim _{x\to \pm \infty }\left}
6670:defines the asymptotes of the curve: setting
3341:Elementary methods for identifying asymptotes
3192:{\displaystyle \lim _{x\to \pm \infty }\left}
3071:tends to +∞, and in the second case the line
397:{\displaystyle \left(x,{\frac {1}{x}}\right)}
150:) which means "not falling together", from ἀ
8316:Cambridge, University Press, 1920, pp 89ff.(
6216:be a parametric plane curve, in coordinates
6002:An important case is when the curve is the
4755:
37:The graph of a function with a horizontal (
8102:
8084:
5676:-axis is an asymptote of the curve. Also,
3531:
3390:
2167:is a horizontal asymptote of the function
1314:This function has a vertical asymptote at
554:
8225:
8116:
7024:{\displaystyle Q'_{x}(b,a)=Q'_{y}(b,a)=0}
6280:is simply referred to as an asymptote of
6163:
4358:{\displaystyle f(x)={\frac {1}{x^{2}+1}}}
3548:
3407:
2956:
8437:L.P. Siceloff, G. Wentworth, D.E. Smith
8366:
7318:
6412:
6256:if the shortest distance from the point
6167:
5451:
5380:of the function also have an asymptote.
4961:
4958:Oblique asymptotes of rational functions
3353:The oblique asymptote, for the function
2761:
2136:
2132:
828:from the left (from lesser values), and
247:
56:
32:
8344:Elementary Geometry of Algebraic Curves
8331:An elementary treatise on curve tracing
8298:An elementary treatise on curve tracing
8171:
8052:
1036:and the curve has a vertical asymptote
142:The word asymptote is derived from the
8500:
8406:Introduction to plane algebraic curves
8248:
7448:{\displaystyle y=\pm {\frac {b}{a}}x.}
7245:has no real points outside the square
6437:are the lines that are tangent to the
5255:. This is because the other term, 1/(
5039:{\displaystyle f(x)=(x^{2}+x+1)/(x+1)}
3132: = 0) as seen in the limits
2809:{\displaystyle f(x)=x+{\tfrac {1}{x}}}
2739:{\displaystyle x\mapsto \exp(-x^{2}),}
613:
6397:Asymptotes are used in procedures of
6121:. The non-vertical case has equation
5447:
2860:) is asymptotic to the straight line
2757:
2145:function has two different asymptotes
2103:doesn't have a vertical asymptote at
1877:{\displaystyle \quad n=0,1,2,\ldots }
1055:, and its precise value at the point
165:There are three kinds of asymptotes:
8403:
8369:A treatise on algebraic plane curves
6460:intersects its asymptote at most at
6445:. For example, one may identify the
6034:)). For this, a parameterization is
5239:shown to the right. As the value of
4536:of the numerator by the denominator
4402:= the ratio of leading coefficients
288:{\displaystyle f(x)={\tfrac {1}{x}}}
7296:{\displaystyle |x|\leq 1,|y|\leq 1}
6408:
495:-axis. Similarly, as the values of
354:{\displaystyle f(x)={\frac {1}{x}}}
319:Consider the graph of the function
13:
7314:
7078:{\displaystyle P_{n-1}(b,a)\neq 0}
6080:{\displaystyle t\mapsto (t,f(t)).}
5582:
5270:Transformations of known functions
4194:
4145:
4075:
4033:
3883:
3834:
3740:
3698:
3494:
3471:
3377:is computed first and is given by
3306:
3224:
3156:
3116:, which has the oblique asymptote
2978:
2902:
2629:
2582:
2440:
2373:
2257:
2203:
2056:
2033:
1618:
1471:
1020:
952:
861:{\displaystyle \lim _{x\to a^{+}}}
813:{\displaystyle \lim _{x\to a^{-}}}
761:
696:
14:
8524:
8465:
8408:, Boston, MA: Birkhäuser Boston,
5747:then the distance from the point
5632:can be defined parametrically as
4240:Asymptotes for rational functions
3361:), will be given by the equation
8367:Coolidge, Julian Lowell (1959),
6447:asymptotes to the unit hyperbola
6179:+3 is a parabolic asymptote to (
4515:{\displaystyle y={\frac {2}{3}}}
4265:deg(numerator)−deg(denominator)
4232:does not have an asymptote when
75:
8445:
8430:
8397:
8360:
8336:
8323:
8306:
8290:
8255:, Cambridge, University Press,
5672: → ∞. Therefore, the
5605:if the distance from the point
4258:
1846:
1828:
1290:
1272:
1051:) may or may not be defined at
243:
217:More generally, one curve is a
8277:
8242:
8219:
8201:
8165:
8152:
8140:
8096:
8078:
8011:{\displaystyle P_{d}(x,y,z)=0}
7999:
7981:
7836:
7818:
7796:
7778:
7283:
7275:
7261:
7253:
7190:
7178:
7156:
7144:
7125:
7113:
7066:
7054:
7012:
7000:
6981:
6969:
6936:
6924:
6891:
6879:
6837:
6825:
6800:
6788:
6766:
6754:
6615:
6603:
6581:
6569:
6547:
6535:
6519:
6507:
6393:Asymptotes and curve sketching
6252:is a curvilinear asymptote of
6071:
6068:
6062:
6050:
6047:
5954:
5944:
5941:
5935:
5929:
5917:
5914:
5908:
5902:
5892:
5829:
5819:
5813:
5801:
5795:
5785:
5576:
5573:
5567:
5551:
5545:
5532:
5524:
5154:
5148:
5033:
5021:
5013:
4988:
4982:
4976:
4938:
4926:
4923:
4911:
4903:
4891:
4888:
4876:
4796:
4790:
4696:
4690:
4554:
4548:
4420:
4414:
4324:
4318:
4188:
4174:
4162:
4156:
4150:
4139:
4069:
4047:
4041:
4027:
3877:
3863:
3851:
3845:
3839:
3828:
3734:
3712:
3706:
3692:
3589:
3577:
3571:
3565:
3557:
3433:
3427:
3416:
3300:
3218:
3175:
3169:
3150:
3018:
3003:
2997:
2991:
2972:
2942:
2927:
2921:
2915:
2896:
2782:
2776:
2730:
2714:
2705:
2623:
2576:
2542:has a horizontal asymptote at
2457:
2451:
2434:
2390:
2384:
2367:
2271:
2265:
2251:
2217:
2211:
2197:
2090:
2084:
1941:
1928:
1816:
1801:
1790:
1780:
1724:
1715:
1700:
1691:
1679:
1673:
1546:
1532:
1526:
1508:
1399:
1385:
1379:
1361:
1269:
1254:
1227:
1221:
1183:) has the vertical asymptote
1082:
1076:
981:
913:
843:
795:
752:
746:
728:
687:
681:
663:
578:. These can be computed using
484:{\displaystyle {\frac {1}{x}}}
335:
329:
267:
261:
1:
8160:History of Mathematics, vol 2
8085:Williamson, Benjamin (1899),
8041:
6389:rather than a straight line.
6248:tends to infinity. The curve
5361:is a horizontal asymptote of
5341:is a horizontal asymptote of
5121:{\displaystyle x=1,2,3,4,5,6}
630:of the graph of the function
6366:has a curvilinear asymptote
5601:A line ℓ is an asymptote of
5482:plane curve, in coordinates
5440:+2 has horizontal asymptote
3627:is the oblique asymptote of
221:of another (as opposed to a
7:
8060:Encyclopedia of Mathematics
8029:
6942:{\displaystyle Q'_{y}(b,a)}
6897:{\displaystyle Q'_{x}(b,a)}
5771:)) to the line is given by
5318:is a vertical asymptote of
5298:is a vertical asymptote of
4754:no linear asymptote, but a
3085:is an oblique asymptote of
3059:is an oblique asymptote of
3045:In the first case the line
1754:For the sequence of points
213:+∞ or −∞.
205:+∞ or −∞.
41: = 0), vertical (
10:
8529:
8348:Cambridge University Press
8183:(2nd ed.), New York:
7215:Over the complex numbers,
6491:is a polynomial of degree
6464:−2 other points, by
6287:For example, the function
3646:For example, the function
1196:a vertical line in general
563:are of curves of the form
177:. For curves given by the
25:
21:Asymptote (disambiguation)
18:
8226:Pogorelov, A. V. (1959),
8147:Oxford English Dictionary
7957:{\displaystyle P_{d-1}=0}
5740:{\displaystyle ax+by+c=0}
5247:approaches the asymptote
1947:{\displaystyle f'(x_{n})}
1303:{\displaystyle \quad x=0}
610:tends to +∞ or −∞.
307:-axis are the asymptotes.
6456:A plane curve of degree
5613:) to ℓ tends to zero as
3500:{\displaystyle +\infty }
3477:{\displaystyle -\infty }
2062:{\displaystyle -\infty }
2039:{\displaystyle +\infty }
464:becomes, its reciprocal
8207:Reference for section:
8149:, second edition, 1989.
8053:Kuptsov, L.P. (2001) ,
7903:homogeneous polynomials
7410:has the two asymptotes
6949:are not both zero. If
6423:the folium of Descartes
6405:seems to be preferred.
6117:, for some real number
5696: → 0. So the
4218:, which does not exist.
2844:-axis, it is called an
2558:because, respectively,
2159:. The horizontal line
555:Asymptotes of functions
8249:Fowler, R. H. (1920),
8012:
7958:
7919:
7895:
7868:
7749:
7639:
7529:
7449:
7401:
7324:
7297:
7206:
7079:
7025:
6943:
6898:
6850:
6635:
6487:) = 0 where
6426:
6381:, which is known as a
6357:
6195:
6164:Curvilinear asymptotes
6153:
6081:
5990:
5865:
5741:
5692:which approaches 0 as
5668:which approaches 0 as
5592:
5457:
5230:
5128:
5122:
5066:
5040:
4948:
4746:
4665:
4664:{\displaystyle y=2x+3}
4629:
4532:= the quotient of the
4516:
4482:
4386:
4359:
4301:
4274:Asymptote for example
4268:Asymptotes in general
4212:
4108:
3954:
3799:
3596:
3501:
3478:
3448:
3331:
3275:
3193:
3036:
2833:
2810:
2740:
2669:
2480:
2413:
2284:
2230:
2146:
2123:
2097:
2063:
2040:
2017:
1997:
1968:
1948:
1907:
1878:
1833:
1744:
1649:
1625:
1481:
1337:
1304:
1277:
1159:
1027:
959:
862:
814:
771:
706:
539:-axis. Thus, both the
529:
509:
485:
458:
438:
418:
398:
355:
308:
289:
62:
54:
8508:Mathematical analysis
8346:, § 12.6 Asymptotes,
8318:online at archive.org
8228:Differential geometry
8185:John Wiley & Sons
8013:
7959:
7920:
7896:
7894:{\displaystyle P_{i}}
7869:
7750:
7640:
7530:
7450:
7402:
7322:
7298:
7207:
7080:
7026:
6944:
6899:
6851:
6636:
6429:The asymptotes of an
6416:
6358:
6171:
6154:
6082:
5991:
5866:
5742:
5593:
5455:
5231:
5123:
5067:
5046:. Red: the asymptote
5041:
4965:
4949:
4756:curvilinear asymptote
4747:
4666:
4630:
4517:
4483:
4387:
4360:
4302:
4213:
4109:
3955:
3800:
3597:
3502:
3479:
3449:
3332:
3276:
3194:
3037:
2811:
2765:
2741:
2670:
2481:
2414:
2285:
2231:
2150:Horizontal asymptotes
2140:
2133:Horizontal asymptotes
2124:
2098:
2096:{\displaystyle f'(x)}
2064:
2041:
2018:
1998:
1969:
1949:
1908:
1879:
1834:
1745:
1650:
1626:
1482:
1338:
1305:
1278:
1168:has a limit of +∞ as
1160:
1028:
960:
863:
824:approaches the value
815:
772:
707:
530:
510:
486:
459:
439:
419:
399:
356:
297:Cartesian coordinates
290:
251:
219:curvilinear asymptote
60:
36:
8404:Kunz, Ernst (2005),
8213:The Penny Cyclopædia
8105:Mathematics Magazine
7968:
7964:. Then the equation
7929:
7909:
7878:
7765:
7658:
7648:is said to have the
7548:
7465:
7417:
7337:
7249:
7097:
7035:
6953:
6908:
6863:
6738:
6501:
6294:
6143:
6102:can be −∞ and
6041:
5885:
5778:
5707:
5513:
5142:
5076:
5050:
4970:
4966:Black: the graph of
4784:
4684:
4640:
4542:
4493:
4408:
4370:
4312:
4285:
4122:
4010:
3974:is the asymptote of
3811:
3675:
3525:
3488:
3465:
3384:
3286:
3204:
3139:
2885:
2832:are both asymptotes.
2770:
2699:
2565:
2546: = 0 when
2423:
2356:
2318:, and in the second
2240:
2186:
2107:
2073:
2050:
2027:
2007:
1982:
1958:
1917:
1891:
1843:
1761:
1662:
1639:
1497:
1350:
1336:{\displaystyle x=0,}
1318:
1287:
1215:
1070:
970:
902:
832:
784:
717:
652:
582:and classified into
519:
499:
468:
448:
428:
408:
365:
323:
255:
19:For other uses, see
8342:C.G. Gibson (1998)
8283:William Nicholson,
8162:Dover (1958) p. 318
8072:Specific references
7143:
7112:
6999:
6968:
6923:
6878:
6859:is an asymptote if
6787:
6753:
6439:projectivized curve
6383:parabolic asymptote
5416:is an asymptote of
5396:is an asymptote of
5259:+1), approaches 0.
5065:{\displaystyle y=x}
4385:{\displaystyle y=0}
4300:{\displaystyle y=0}
4261:
3518:can be computed by
3514:then the value for
3097:tends to −∞.
2878: ≠ 0) if
2349:function satisfies
2334:as an asymptote as
2294:In the first case,
2122:{\displaystyle x=0}
1906:{\displaystyle x=0}
614:Vertical asymptotes
238:asymptotic analysis
156:Apollonius of Perga
129:projective geometry
8487:2012-02-15 at the
8394:, pp. 40–44.
8373:Dover Publications
8047:General references
8008:
7954:
7915:
7891:
7864:
7745:
7635:
7525:
7445:
7397:
7325:
7293:
7202:
7131:
7100:
7075:
7021:
6987:
6956:
6939:
6911:
6894:
6866:
6846:
6775:
6741:
6631:
6427:
6353:
6196:
6149:
6077:
5986:
5861:
5737:
5680: → ∞ as
5656: → ∞ as
5588:
5531:
5458:
5448:General definition
5226:
5129:
5118:
5062:
5036:
4944:
4742:
4661:
4625:
4534:Euclidean division
4512:
4478:
4382:
4355:
4297:
4259:
4208:
4198:
4149:
4104:
4079:
4037:
3950:
3887:
3838:
3795:
3744:
3702:
3592:
3564:
3497:
3474:
3444:
3423:
3327:
3310:
3271:
3228:
3189:
3160:
3032:
2982:
2962:
2906:
2834:
2824:= 0) and the line
2806:
2804:
2758:Oblique asymptotes
2736:
2692:as it graph), the
2665:
2633:
2586:
2476:
2444:
2409:
2377:
2310:as asymptote when
2280:
2261:
2226:
2207:
2147:
2119:
2093:
2059:
2036:
2013:
1996:{\displaystyle f'}
1993:
1974:. Therefore, both
1964:
1944:
1903:
1874:
1829:
1740:
1713:
1645:
1635:The derivative of
1621:
1601:
1576:
1560:
1522:
1477:
1454:
1429:
1413:
1375:
1333:
1300:
1273:
1267:
1243:
1155:
1150:
1023:
995:
955:
927:
879:For example, if ƒ(
858:
857:
810:
809:
767:
742:
702:
677:
628:vertical asymptote
525:
505:
481:
454:
434:
414:
394:
351:
309:
285:
283:
135:to the curve at a
63:
55:
8513:Analytic geometry
8439:Analytic geometry
8415:978-0-8176-4381-2
8261:2027/uc1.b4073882
8194:978-0-471-00005-1
7918:{\displaystyle i}
7737:
7710:
7683:
7627:
7600:
7573:
7517:
7490:
7437:
7389:
7362:
6443:point at infinity
6351:
6268:tends to zero as
6152:{\displaystyle n}
5984:
5983:
5859:
5858:
5516:
5224:
5197:
4942:
4868:
4762:
4761:
4740:
4623:
4595:
4510:
4476:
4353:
4246:rational function
4180:
4131:
4096:
4061:
4019:
3928:
3869:
3820:
3787:
3726:
3684:
3613:exist. Otherwise
3549:
3545:
3543:
3408:
3404:
3402:
3319:
3292:
3253:
3210:
3142:
2964:
2961:
2888:
2846:oblique asymptote
2803:
2752:logistic function
2694:Gaussian function
2657:
2615:
2610:
2568:
2471:
2426:
2407:
2359:
2345:For example, the
2243:
2189:
2016:{\displaystyle 0}
1967:{\displaystyle 0}
1887:that approaches
1823:
1738:
1712:
1648:{\displaystyle f}
1600:
1575:
1538:
1500:
1453:
1428:
1391:
1353:
1266:
1242:
1137:
1111:
1104:
1012:
973:
944:
905:
895:approaches 1. So
835:
787:
720:
655:
528:{\displaystyle y}
508:{\displaystyle x}
479:
457:{\displaystyle x}
437:{\displaystyle y}
417:{\displaystyle x}
387:
349:
282:
137:point at infinity
125:tends to infinity
67:analytic geometry
8520:
8459:
8449:
8443:
8434:
8428:
8426:
8401:
8395:
8393:
8364:
8358:
8340:
8334:
8327:
8321:
8310:
8304:
8294:
8288:
8281:
8275:
8273:
8246:
8240:
8238:
8223:
8217:
8205:
8199:
8197:
8182:
8169:
8163:
8156:
8150:
8144:
8138:
8137:
8120:
8100:
8094:
8093:
8082:
8067:
8017:
8015:
8014:
8009:
7980:
7979:
7963:
7961:
7960:
7955:
7947:
7946:
7924:
7922:
7921:
7916:
7900:
7898:
7897:
7892:
7890:
7889:
7873:
7871:
7870:
7865:
7854:
7853:
7817:
7816:
7777:
7776:
7754:
7752:
7751:
7746:
7738:
7736:
7735:
7726:
7725:
7716:
7711:
7709:
7708:
7699:
7698:
7689:
7684:
7682:
7681:
7672:
7671:
7662:
7644:
7642:
7641:
7636:
7628:
7626:
7625:
7616:
7615:
7606:
7601:
7599:
7598:
7589:
7588:
7579:
7574:
7572:
7571:
7562:
7561:
7552:
7534:
7532:
7531:
7526:
7518:
7516:
7515:
7506:
7505:
7496:
7491:
7489:
7488:
7479:
7478:
7469:
7454:
7452:
7451:
7446:
7438:
7430:
7406:
7404:
7403:
7398:
7390:
7388:
7387:
7378:
7377:
7368:
7363:
7361:
7360:
7351:
7350:
7341:
7302:
7300:
7299:
7294:
7286:
7278:
7264:
7256:
7244:
7211:
7209:
7208:
7203:
7177:
7176:
7139:
7108:
7091:
7090:
7089:parabolic branch
7084:
7082:
7081:
7076:
7053:
7052:
7030:
7028:
7027:
7022:
6995:
6964:
6948:
6946:
6945:
6940:
6919:
6903:
6901:
6900:
6895:
6874:
6855:
6853:
6852:
6847:
6824:
6823:
6783:
6749:
6731:, then the line
6730:
6684:
6640:
6638:
6637:
6632:
6630:
6629:
6602:
6601:
6568:
6567:
6534:
6533:
6466:Bézout's theorem
6409:Algebraic curves
6403:asymptotic curve
6385:because it is a
6380:
6362:
6360:
6359:
6354:
6352:
6347:
6331:
6330:
6315:
6314:
6304:
6264:) to a point on
6215:
6158:
6156:
6155:
6150:
6134:
6086:
6084:
6083:
6078:
5995:
5993:
5992:
5987:
5985:
5982:
5981:
5969:
5968:
5959:
5958:
5957:
5895:
5889:
5870:
5868:
5867:
5862:
5860:
5857:
5856:
5844:
5843:
5834:
5833:
5832:
5788:
5782:
5746:
5744:
5743:
5738:
5652:> 0). First,
5597:
5595:
5594:
5589:
5566:
5565:
5544:
5543:
5530:
5477:
5235:
5233:
5232:
5227:
5225:
5223:
5209:
5198:
5196:
5185:
5172:
5171:
5161:
5127:
5125:
5124:
5119:
5071:
5069:
5068:
5063:
5045:
5043:
5042:
5037:
5020:
5000:
4999:
4953:
4951:
4950:
4945:
4943:
4941:
4906:
4874:
4869:
4867:
4857:
4856:
4841:
4840:
4830:
4814:
4813:
4803:
4773:= 1, but not at
4751:
4749:
4748:
4743:
4741:
4739:
4732:
4731:
4718:
4717:
4716:
4703:
4670:
4668:
4667:
4662:
4634:
4632:
4631:
4626:
4624:
4616:
4596:
4591:
4575:
4574:
4561:
4521:
4519:
4518:
4513:
4511:
4503:
4487:
4485:
4484:
4479:
4477:
4475:
4462:
4461:
4448:
4441:
4440:
4427:
4391:
4389:
4388:
4383:
4364:
4362:
4361:
4356:
4354:
4352:
4345:
4344:
4331:
4306:
4304:
4303:
4298:
4262:
4231:
4217:
4215:
4214:
4209:
4197:
4148:
4113:
4111:
4110:
4105:
4097:
4092:
4081:
4078:
4054:
4036:
4002:
3973:
3959:
3957:
3956:
3951:
3943:
3939:
3929:
3924:
3908:
3907:
3894:
3886:
3837:
3804:
3802:
3801:
3796:
3788:
3786:
3785:
3776:
3760:
3759:
3746:
3743:
3719:
3701:
3667:
3626:
3601:
3599:
3598:
3593:
3563:
3547:
3546:
3544:
3541:
3539:
3534:
3506:
3504:
3503:
3498:
3483:
3481:
3480:
3475:
3453:
3451:
3450:
3445:
3440:
3422:
3406:
3405:
3403:
3400:
3398:
3393:
3373:. The value for
3336:
3334:
3333:
3328:
3320:
3312:
3309:
3280:
3278:
3277:
3272:
3270:
3266:
3259:
3255:
3254:
3246:
3227:
3198:
3196:
3195:
3190:
3188:
3184:
3159:
3128: = 1,
3084:
3058:
3041:
3039:
3038:
3033:
3025:
3021:
2981:
2963:
2959:
2949:
2945:
2905:
2873:
2815:
2813:
2812:
2807:
2805:
2796:
2766:In the graph of
2745:
2743:
2742:
2737:
2729:
2728:
2687:
2674:
2672:
2671:
2666:
2658:
2656:
2649:
2648:
2635:
2632:
2611:
2609:
2602:
2601:
2588:
2585:
2557:
2553:
2541:
2526:
2518:
2516:
2507:
2499:
2497:
2485:
2483:
2482:
2477:
2472:
2464:
2443:
2418:
2416:
2415:
2410:
2408:
2400:
2376:
2341:
2317:
2289:
2287:
2286:
2281:
2260:
2235:
2233:
2232:
2227:
2206:
2158:
2157:→ ±∞
2128:
2126:
2125:
2120:
2102:
2100:
2099:
2094:
2083:
2068:
2066:
2065:
2060:
2045:
2043:
2042:
2037:
2022:
2020:
2019:
2014:
2002:
2000:
1999:
1994:
1992:
1976:one-sided limits
1973:
1971:
1970:
1965:
1953:
1951:
1950:
1945:
1940:
1939:
1927:
1912:
1910:
1909:
1904:
1883:
1881:
1880:
1875:
1838:
1836:
1835:
1830:
1824:
1822:
1799:
1798:
1797:
1778:
1773:
1772:
1749:
1747:
1746:
1741:
1739:
1737:
1736:
1727:
1714:
1705:
1686:
1672:
1655:is the function
1654:
1652:
1651:
1646:
1630:
1628:
1627:
1622:
1611:
1607:
1606:
1602:
1593:
1577:
1568:
1559:
1558:
1557:
1521:
1520:
1519:
1486:
1484:
1483:
1478:
1464:
1460:
1459:
1455:
1446:
1430:
1421:
1412:
1411:
1410:
1374:
1373:
1372:
1342:
1340:
1339:
1334:
1309:
1307:
1306:
1301:
1282:
1280:
1279:
1274:
1268:
1259:
1244:
1235:
1189:
1174:
1164:
1162:
1161:
1156:
1154:
1153:
1138:
1135:
1112:
1109:
1105:
1097:
1032:
1030:
1029:
1024:
1013:
1011:
997:
994:
993:
992:
964:
962:
961:
956:
945:
943:
929:
926:
925:
924:
876:from the right.
868:is the limit as
867:
865:
864:
859:
856:
855:
854:
820:is the limit as
819:
817:
816:
811:
808:
807:
806:
776:
774:
773:
768:
741:
740:
739:
711:
709:
708:
703:
676:
675:
674:
644:
577:
534:
532:
531:
526:
514:
512:
511:
506:
490:
488:
487:
482:
480:
472:
463:
461:
460:
455:
443:
441:
440:
435:
423:
421:
420:
415:
403:
401:
400:
395:
393:
389:
388:
380:
360:
358:
357:
352:
350:
342:
294:
292:
291:
286:
284:
275:
223:linear asymptote
214:
206:
198:
110:
109:
106:
105:
102:
99:
96:
93:
90:
87:
84:
81:
8528:
8527:
8523:
8522:
8521:
8519:
8518:
8517:
8498:
8497:
8489:Wayback Machine
8468:
8463:
8462:
8450:
8446:
8435:
8431:
8416:
8402:
8398:
8383:
8365:
8361:
8341:
8337:
8328:
8324:
8311:
8307:
8295:
8291:
8282:
8278:
8271:
8247:
8243:
8224:
8220:
8206:
8202:
8195:
8173:Apostol, Tom M.
8170:
8166:
8157:
8153:
8145:
8141:
8127:10.2307/2690881
8101:
8097:
8083:
8079:
8044:
8032:
8024:asymptotic cone
7975:
7971:
7969:
7966:
7965:
7936:
7932:
7930:
7927:
7926:
7910:
7907:
7906:
7885:
7881:
7879:
7876:
7875:
7849:
7845:
7806:
7802:
7772:
7768:
7766:
7763:
7762:
7731:
7727:
7721:
7717:
7715:
7704:
7700:
7694:
7690:
7688:
7677:
7673:
7667:
7663:
7661:
7659:
7656:
7655:
7650:asymptotic cone
7621:
7617:
7611:
7607:
7605:
7594:
7590:
7584:
7580:
7578:
7567:
7563:
7557:
7553:
7551:
7549:
7546:
7545:
7538:Similarly, the
7511:
7507:
7501:
7497:
7495:
7484:
7480:
7474:
7470:
7468:
7466:
7463:
7462:
7429:
7418:
7415:
7414:
7383:
7379:
7373:
7369:
7367:
7356:
7352:
7346:
7342:
7340:
7338:
7335:
7334:
7317:
7315:Asymptotic cone
7282:
7274:
7260:
7252:
7250:
7247:
7246:
7235:
7232:
7223:
7166:
7162:
7135:
7104:
7098:
7095:
7094:
7088:
7087:
7042:
7038:
7036:
7033:
7032:
6991:
6960:
6954:
6951:
6950:
6915:
6909:
6906:
6905:
6870:
6864:
6861:
6860:
6813:
6809:
6779:
6745:
6739:
6736:
6735:
6720:
6694:
6686:
6683:
6671:
6669:
6652:
6625:
6621:
6597:
6593:
6557:
6553:
6529:
6525:
6502:
6499:
6498:
6431:algebraic curve
6411:
6399:curve sketching
6395:
6367:
6326:
6322:
6310:
6306:
6305:
6303:
6295:
6292:
6291:
6224:) = (
6199:
6166:
6144:
6141:
6140:
6122:
6042:
6039:
6038:
5977:
5973:
5964:
5960:
5953:
5891:
5890:
5888:
5886:
5883:
5882:
5852:
5848:
5839:
5835:
5828:
5784:
5783:
5781:
5779:
5776:
5775:
5755:) = (
5708:
5705:
5704:
5644: = 1/
5628: = 1/
5561:
5557:
5539:
5535:
5520:
5514:
5511:
5510:
5490:) = (
5461:
5450:
5272:
5213:
5208:
5186:
5167:
5163:
5162:
5160:
5143:
5140:
5139:
5077:
5074:
5073:
5051:
5048:
5047:
5016:
4995:
4991:
4971:
4968:
4967:
4960:
4907:
4875:
4873:
4852:
4848:
4836:
4832:
4831:
4809:
4805:
4804:
4802:
4785:
4782:
4781:
4727:
4723:
4719:
4712:
4708:
4704:
4702:
4685:
4682:
4681:
4641:
4638:
4637:
4615:
4570:
4566:
4562:
4560:
4543:
4540:
4539:
4502:
4494:
4491:
4490:
4457:
4453:
4449:
4436:
4432:
4428:
4426:
4409:
4406:
4405:
4371:
4368:
4367:
4340:
4336:
4335:
4330:
4313:
4310:
4309:
4286:
4283:
4282:
4242:
4223:
4184:
4135:
4123:
4120:
4119:
4082:
4080:
4065:
4050:
4023:
4011:
4008:
4007:
3990:
3964:
3903:
3899:
3895:
3893:
3892:
3888:
3873:
3824:
3812:
3809:
3808:
3781:
3777:
3755:
3751:
3747:
3745:
3730:
3715:
3688:
3676:
3673:
3672:
3647:
3614:
3553:
3540:
3535:
3533:
3532:
3526:
3523:
3522:
3489:
3486:
3485:
3466:
3463:
3462:
3436:
3412:
3399:
3394:
3392:
3391:
3385:
3382:
3381:
3351:
3343:
3311:
3296:
3287:
3284:
3283:
3245:
3238:
3234:
3233:
3229:
3214:
3205:
3202:
3201:
3165:
3161:
3146:
3140:
3137:
3136:
3072:
3046:
2987:
2983:
2968:
2957:
2911:
2907:
2892:
2886:
2883:
2882:
2861:
2850:slant asymptote
2794:
2771:
2768:
2767:
2760:
2724:
2720:
2700:
2697:
2696:
2679:
2644:
2640:
2639:
2634:
2619:
2597:
2593:
2592:
2587:
2572:
2566:
2563:
2562:
2555:
2551:
2531:
2524:
2514:
2509:
2505:
2495:
2490:
2463:
2430:
2424:
2421:
2420:
2399:
2363:
2357:
2354:
2353:
2339:
2315:
2247:
2241:
2238:
2237:
2193:
2187:
2184:
2183:
2153:
2135:
2108:
2105:
2104:
2076:
2074:
2071:
2070:
2051:
2048:
2047:
2028:
2025:
2024:
2023:can be neither
2008:
2005:
2004:
1985:
1983:
1980:
1979:
1959:
1956:
1955:
1954:are constantly
1935:
1931:
1920:
1918:
1915:
1914:
1892:
1889:
1888:
1844:
1841:
1840:
1800:
1793:
1789:
1779:
1777:
1768:
1764:
1762:
1759:
1758:
1732:
1728:
1703:
1687:
1685:
1665:
1663:
1660:
1659:
1640:
1637:
1636:
1591:
1587:
1566:
1565:
1561:
1553:
1549:
1542:
1515:
1511:
1504:
1498:
1495:
1494:
1444:
1440:
1419:
1418:
1414:
1406:
1402:
1395:
1368:
1364:
1357:
1351:
1348:
1347:
1319:
1316:
1315:
1288:
1285:
1284:
1257:
1233:
1216:
1213:
1212:
1184:
1169:
1149:
1148:
1134:
1132:
1126:
1125:
1108:
1106:
1096:
1089:
1088:
1071:
1068:
1067:
1001:
996:
988:
984:
977:
971:
968:
967:
933:
928:
920:
916:
909:
903:
900:
899:
850:
846:
839:
833:
830:
829:
802:
798:
791:
785:
782:
781:
735:
731:
724:
718:
715:
714:
670:
666:
659:
653:
650:
649:
631:
616:
564:
557:
520:
517:
516:
500:
497:
496:
471:
469:
466:
465:
449:
446:
445:
429:
426:
425:
409:
406:
405:
379:
372:
368:
366:
363:
362:
341:
324:
321:
320:
273:
256:
253:
252:
246:
212:
204:
185:
158:in his work on
78:
74:
31:
24:
17:
12:
11:
5:
8526:
8516:
8515:
8510:
8496:
8495:
8493:Science Museum
8479:
8467:
8466:External links
8464:
8461:
8460:
8454:Solid geometry
8444:
8429:
8414:
8396:
8381:
8359:
8335:
8333:, 1918, page 5
8322:
8312:Fowler, R. H.
8305:
8289:
8287:, Vol. 5, 1809
8276:
8269:
8241:
8218:
8200:
8193:
8164:
8151:
8139:
8111:(3): 183–192,
8095:
8076:
8075:
8074:
8073:
8069:
8068:
8049:
8048:
8043:
8040:
8039:
8038:
8036:Big O notation
8031:
8028:
8007:
8004:
8001:
7998:
7995:
7992:
7989:
7986:
7983:
7978:
7974:
7953:
7950:
7945:
7942:
7939:
7935:
7914:
7888:
7884:
7863:
7860:
7857:
7852:
7848:
7844:
7841:
7838:
7835:
7832:
7829:
7826:
7823:
7820:
7815:
7812:
7809:
7805:
7801:
7798:
7795:
7792:
7789:
7786:
7783:
7780:
7775:
7771:
7756:
7755:
7744:
7741:
7734:
7730:
7724:
7720:
7714:
7707:
7703:
7697:
7693:
7687:
7680:
7676:
7670:
7666:
7646:
7645:
7634:
7631:
7624:
7620:
7614:
7610:
7604:
7597:
7593:
7587:
7583:
7577:
7570:
7566:
7560:
7556:
7536:
7535:
7524:
7521:
7514:
7510:
7504:
7500:
7494:
7487:
7483:
7477:
7473:
7456:
7455:
7444:
7441:
7436:
7433:
7428:
7425:
7422:
7408:
7407:
7396:
7393:
7386:
7382:
7376:
7372:
7366:
7359:
7355:
7349:
7345:
7316:
7313:
7292:
7289:
7285:
7281:
7277:
7273:
7270:
7267:
7263:
7259:
7255:
7228:
7219:
7201:
7198:
7195:
7192:
7189:
7186:
7183:
7180:
7175:
7172:
7169:
7165:
7161:
7158:
7155:
7152:
7149:
7146:
7142:
7138:
7134:
7130:
7127:
7124:
7121:
7118:
7115:
7111:
7107:
7103:
7074:
7071:
7068:
7065:
7062:
7059:
7056:
7051:
7048:
7045:
7041:
7020:
7017:
7014:
7011:
7008:
7005:
7002:
6998:
6994:
6990:
6986:
6983:
6980:
6977:
6974:
6971:
6967:
6963:
6959:
6938:
6935:
6932:
6929:
6926:
6922:
6918:
6914:
6893:
6890:
6887:
6884:
6881:
6877:
6873:
6869:
6857:
6856:
6845:
6842:
6839:
6836:
6833:
6830:
6827:
6822:
6819:
6816:
6812:
6808:
6805:
6802:
6799:
6796:
6793:
6790:
6786:
6782:
6778:
6774:
6771:
6768:
6765:
6762:
6759:
6756:
6752:
6748:
6744:
6715:
6690:
6679:
6665:
6648:
6642:
6641:
6628:
6624:
6620:
6617:
6614:
6611:
6608:
6605:
6600:
6596:
6592:
6589:
6586:
6583:
6580:
6577:
6574:
6571:
6566:
6563:
6560:
6556:
6552:
6549:
6546:
6543:
6540:
6537:
6532:
6528:
6524:
6521:
6518:
6515:
6512:
6509:
6506:
6410:
6407:
6394:
6391:
6364:
6363:
6350:
6346:
6343:
6340:
6337:
6334:
6329:
6325:
6321:
6318:
6313:
6309:
6302:
6299:
6165:
6162:
6148:
6088:
6087:
6076:
6073:
6070:
6067:
6064:
6061:
6058:
6055:
6052:
6049:
6046:
5997:
5996:
5980:
5976:
5972:
5967:
5963:
5956:
5952:
5949:
5946:
5943:
5940:
5937:
5934:
5931:
5928:
5925:
5922:
5919:
5916:
5913:
5910:
5907:
5904:
5901:
5898:
5894:
5872:
5871:
5855:
5851:
5847:
5842:
5838:
5831:
5827:
5824:
5821:
5818:
5815:
5812:
5809:
5806:
5803:
5800:
5797:
5794:
5791:
5787:
5736:
5733:
5730:
5727:
5724:
5721:
5718:
5715:
5712:
5599:
5598:
5587:
5584:
5581:
5578:
5575:
5572:
5569:
5564:
5560:
5556:
5553:
5550:
5547:
5542:
5538:
5534:
5529:
5526:
5523:
5519:
5449:
5446:
5426:
5425:
5374:
5373:
5331:
5271:
5268:
5237:
5236:
5222:
5219:
5216:
5212:
5207:
5204:
5201:
5195:
5192:
5189:
5184:
5181:
5178:
5175:
5170:
5166:
5159:
5156:
5153:
5150:
5147:
5117:
5114:
5111:
5108:
5105:
5102:
5099:
5096:
5093:
5090:
5087:
5084:
5081:
5061:
5058:
5055:
5035:
5032:
5029:
5026:
5023:
5019:
5015:
5012:
5009:
5006:
5003:
4998:
4994:
4990:
4987:
4984:
4981:
4978:
4975:
4959:
4956:
4955:
4954:
4940:
4937:
4934:
4931:
4928:
4925:
4922:
4919:
4916:
4913:
4910:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4881:
4878:
4872:
4866:
4863:
4860:
4855:
4851:
4847:
4844:
4839:
4835:
4829:
4826:
4823:
4820:
4817:
4812:
4808:
4801:
4798:
4795:
4792:
4789:
4760:
4759:
4752:
4738:
4735:
4730:
4726:
4722:
4715:
4711:
4707:
4701:
4698:
4695:
4692:
4689:
4679:
4676:
4672:
4671:
4660:
4657:
4654:
4651:
4648:
4645:
4635:
4622:
4619:
4614:
4611:
4608:
4605:
4602:
4599:
4594:
4590:
4587:
4584:
4581:
4578:
4573:
4569:
4565:
4559:
4556:
4553:
4550:
4547:
4537:
4527:
4523:
4522:
4509:
4506:
4501:
4498:
4488:
4474:
4471:
4468:
4465:
4460:
4456:
4452:
4447:
4444:
4439:
4435:
4431:
4425:
4422:
4419:
4416:
4413:
4403:
4397:
4393:
4392:
4381:
4378:
4375:
4365:
4351:
4348:
4343:
4339:
4334:
4329:
4326:
4323:
4320:
4317:
4307:
4296:
4293:
4290:
4280:
4276:
4275:
4272:
4269:
4266:
4241:
4238:
4220:
4219:
4207:
4204:
4201:
4196:
4193:
4190:
4187:
4183:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4147:
4144:
4141:
4138:
4134:
4130:
4127:
4116:
4115:
4103:
4100:
4095:
4091:
4088:
4085:
4077:
4074:
4071:
4068:
4064:
4060:
4057:
4053:
4049:
4046:
4043:
4040:
4035:
4032:
4029:
4026:
4022:
4018:
4015:
3961:
3960:
3949:
3946:
3942:
3938:
3935:
3932:
3927:
3923:
3920:
3917:
3914:
3911:
3906:
3902:
3898:
3891:
3885:
3882:
3879:
3876:
3872:
3868:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3844:
3841:
3836:
3833:
3830:
3827:
3823:
3819:
3816:
3806:
3794:
3791:
3784:
3780:
3775:
3772:
3769:
3766:
3763:
3758:
3754:
3750:
3742:
3739:
3736:
3733:
3729:
3725:
3722:
3718:
3714:
3711:
3708:
3705:
3700:
3697:
3694:
3691:
3687:
3683:
3680:
3603:
3602:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3562:
3559:
3556:
3552:
3538:
3530:
3496:
3493:
3473:
3470:
3455:
3454:
3443:
3439:
3435:
3432:
3429:
3426:
3421:
3418:
3415:
3411:
3397:
3389:
3350:
3347:
3342:
3339:
3338:
3337:
3326:
3323:
3318:
3315:
3308:
3305:
3302:
3299:
3295:
3291:
3281:
3269:
3265:
3262:
3258:
3252:
3249:
3244:
3241:
3237:
3232:
3226:
3223:
3220:
3217:
3213:
3209:
3199:
3187:
3183:
3180:
3177:
3174:
3171:
3168:
3164:
3158:
3155:
3152:
3149:
3145:
3108:) =
3100:An example is
3043:
3042:
3031:
3028:
3024:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2986:
2980:
2977:
2974:
2971:
2967:
2960: or
2955:
2952:
2948:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2910:
2904:
2901:
2898:
2895:
2891:
2802:
2799:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2759:
2756:
2748:error function
2735:
2732:
2727:
2723:
2719:
2716:
2713:
2710:
2707:
2704:
2676:
2675:
2664:
2661:
2655:
2652:
2647:
2643:
2638:
2631:
2628:
2625:
2622:
2618:
2614:
2608:
2605:
2600:
2596:
2591:
2584:
2581:
2578:
2575:
2571:
2550:tends both to
2487:
2486:
2475:
2470:
2467:
2462:
2459:
2456:
2453:
2450:
2447:
2442:
2439:
2436:
2433:
2429:
2406:
2403:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2375:
2372:
2369:
2366:
2362:
2292:
2291:
2279:
2276:
2273:
2270:
2267:
2264:
2259:
2256:
2253:
2250:
2246:
2225:
2222:
2219:
2216:
2213:
2210:
2205:
2202:
2199:
2196:
2192:
2134:
2131:
2118:
2115:
2112:
2092:
2089:
2086:
2082:
2079:
2058:
2055:
2035:
2032:
2012:
1991:
1988:
1963:
1943:
1938:
1934:
1930:
1926:
1923:
1902:
1899:
1896:
1885:
1884:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1827:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1796:
1792:
1788:
1785:
1782:
1776:
1771:
1767:
1752:
1751:
1735:
1731:
1726:
1723:
1720:
1717:
1711:
1708:
1702:
1699:
1696:
1693:
1690:
1684:
1681:
1678:
1675:
1671:
1668:
1644:
1633:
1632:
1620:
1617:
1614:
1610:
1605:
1599:
1596:
1590:
1586:
1583:
1580:
1574:
1571:
1564:
1556:
1552:
1548:
1545:
1541:
1537:
1534:
1531:
1528:
1525:
1518:
1514:
1510:
1507:
1503:
1488:
1487:
1476:
1473:
1470:
1467:
1463:
1458:
1452:
1449:
1443:
1439:
1436:
1433:
1427:
1424:
1417:
1409:
1405:
1401:
1398:
1394:
1390:
1387:
1384:
1381:
1378:
1371:
1367:
1363:
1360:
1356:
1332:
1329:
1326:
1323:
1312:
1311:
1299:
1296:
1293:
1271:
1265:
1262:
1256:
1253:
1250:
1247:
1241:
1238:
1232:
1229:
1226:
1223:
1220:
1190:, even though
1166:
1165:
1152:
1147:
1144:
1141:
1133:
1131:
1128:
1127:
1124:
1121:
1118:
1115:
1107:
1103:
1100:
1095:
1094:
1092:
1087:
1084:
1081:
1078:
1075:
1034:
1033:
1022:
1019:
1016:
1010:
1007:
1004:
1000:
991:
987:
983:
980:
976:
965:
954:
951:
948:
942:
939:
936:
932:
923:
919:
915:
912:
908:
853:
849:
845:
842:
838:
805:
801:
797:
794:
790:
778:
777:
766:
763:
760:
757:
754:
751:
748:
745:
738:
734:
730:
727:
723:
712:
701:
698:
695:
692:
689:
686:
683:
680:
673:
669:
665:
662:
658:
615:
612:
556:
553:
524:
504:
478:
475:
453:
433:
413:
392:
386:
383:
378:
375:
371:
348:
345:
340:
337:
334:
331:
328:
281:
278:
272:
269:
266:
263:
260:
245:
242:
160:conic sections
49: = 2
15:
9:
6:
4:
3:
2:
8525:
8514:
8511:
8509:
8506:
8505:
8503:
8494:
8490:
8486:
8483:
8480:
8477:
8473:
8470:
8469:
8457:
8455:
8448:
8442:
8441:(1922) p. 271
8440:
8433:
8425:
8421:
8417:
8411:
8407:
8400:
8392:
8388:
8384:
8382:0-486-49576-0
8378:
8374:
8370:
8363:
8356:
8355:0-521-64140-3
8352:
8349:
8345:
8339:
8332:
8326:
8319:
8315:
8309:
8303:
8299:
8293:
8286:
8280:
8272:
8270:0-486-44277-2
8266:
8262:
8258:
8254:
8253:
8245:
8237:
8233:
8229:
8222:
8215:
8214:
8210:
8204:
8196:
8190:
8186:
8181:
8180:
8174:
8168:
8161:
8155:
8148:
8143:
8136:
8132:
8128:
8124:
8119:
8118:10.1.1.502.72
8114:
8110:
8106:
8099:
8092:
8088:
8081:
8077:
8071:
8070:
8066:
8062:
8061:
8056:
8051:
8050:
8046:
8045:
8037:
8034:
8033:
8027:
8025:
8021:
8005:
8002:
7996:
7993:
7990:
7987:
7984:
7976:
7972:
7951:
7948:
7943:
7940:
7937:
7933:
7912:
7904:
7886:
7882:
7861:
7858:
7855:
7850:
7846:
7842:
7839:
7833:
7830:
7827:
7824:
7821:
7813:
7810:
7807:
7803:
7799:
7793:
7790:
7787:
7784:
7781:
7773:
7769:
7759:
7742:
7739:
7732:
7728:
7722:
7718:
7712:
7705:
7701:
7695:
7691:
7685:
7678:
7674:
7668:
7664:
7654:
7653:
7652:
7651:
7632:
7629:
7622:
7618:
7612:
7608:
7602:
7595:
7591:
7585:
7581:
7575:
7568:
7564:
7558:
7554:
7544:
7543:
7542:
7541:
7522:
7519:
7512:
7508:
7502:
7498:
7492:
7485:
7481:
7475:
7471:
7461:
7460:
7459:
7442:
7439:
7434:
7431:
7426:
7423:
7420:
7413:
7412:
7411:
7394:
7391:
7384:
7380:
7374:
7370:
7364:
7357:
7353:
7347:
7343:
7333:
7332:
7331:
7330:
7321:
7312:
7310:
7306:
7290:
7287:
7279:
7271:
7268:
7265:
7257:
7242:
7238:
7231:
7227:
7222:
7218:
7213:
7199:
7196:
7193:
7187:
7184:
7181:
7173:
7170:
7167:
7163:
7159:
7153:
7150:
7147:
7140:
7136:
7132:
7128:
7122:
7119:
7116:
7109:
7105:
7101:
7092:
7072:
7069:
7063:
7060:
7057:
7049:
7046:
7043:
7039:
7018:
7015:
7009:
7006:
7003:
6996:
6992:
6988:
6984:
6978:
6975:
6972:
6965:
6961:
6957:
6933:
6930:
6927:
6920:
6916:
6912:
6888:
6885:
6882:
6875:
6871:
6867:
6843:
6840:
6834:
6831:
6828:
6820:
6817:
6814:
6810:
6806:
6803:
6797:
6794:
6791:
6784:
6780:
6776:
6772:
6769:
6763:
6760:
6757:
6750:
6746:
6742:
6734:
6733:
6732:
6728:
6724:
6718:
6714:
6710:
6706:
6702:
6698:
6693:
6689:
6682:
6678:
6674:
6668:
6664:
6660:
6656:
6651:
6647:
6626:
6622:
6618:
6612:
6609:
6606:
6598:
6594:
6590:
6587:
6584:
6578:
6575:
6572:
6564:
6561:
6558:
6554:
6550:
6544:
6541:
6538:
6530:
6526:
6522:
6516:
6513:
6510:
6504:
6497:
6496:
6495:
6494:
6490:
6486:
6482:
6478:
6473:
6471:
6467:
6463:
6459:
6454:
6452:
6448:
6444:
6440:
6436:
6432:
6424:
6420:
6415:
6406:
6404:
6400:
6390:
6388:
6384:
6378:
6374:
6370:
6348:
6344:
6341:
6338:
6335:
6332:
6327:
6323:
6319:
6316:
6311:
6307:
6300:
6297:
6290:
6289:
6288:
6285:
6283:
6279:
6276:. Sometimes
6275:
6272: →
6271:
6267:
6263:
6259:
6255:
6251:
6247:
6243:
6239:
6235:
6231:
6227:
6223:
6219:
6214:
6210:
6206:
6202:
6194:
6190:
6186:
6182:
6178:
6174:
6170:
6161:
6146:
6138:
6133:
6129:
6125:
6120:
6116:
6113: =
6112:
6107:
6105:
6101:
6097:
6093:
6074:
6065:
6059:
6056:
6053:
6044:
6037:
6036:
6035:
6033:
6029:
6025:
6021:
6017:
6014: =
6013:
6009:
6008:real function
6005:
6000:
5978:
5974:
5970:
5965:
5961:
5950:
5947:
5938:
5932:
5926:
5923:
5920:
5911:
5905:
5899:
5896:
5881:
5880:
5879:
5877:
5853:
5849:
5845:
5840:
5836:
5825:
5822:
5816:
5810:
5807:
5804:
5798:
5792:
5789:
5774:
5773:
5772:
5770:
5766:
5762:
5758:
5754:
5750:
5734:
5731:
5728:
5725:
5722:
5719:
5716:
5713:
5710:
5701:
5699:
5695:
5691:
5687:
5683:
5679:
5675:
5671:
5667:
5663:
5659:
5655:
5651:
5647:
5643:
5639:
5636: =
5635:
5631:
5627:
5622:
5620:
5617: →
5616:
5612:
5608:
5604:
5585:
5579:
5570:
5562:
5558:
5554:
5548:
5540:
5536:
5527:
5521:
5509:
5508:
5507:
5505:
5501:
5497:
5493:
5489:
5485:
5481:
5476:
5472:
5468:
5464:
5454:
5445:
5443:
5439:
5435:
5431:
5428:For example,
5423:
5419:
5415:
5411:
5407:
5403:
5399:
5395:
5391:
5387:
5383:
5382:
5381:
5379:
5372:
5368:
5364:
5360:
5356:
5352:
5348:
5344:
5340:
5336:
5332:
5329:
5325:
5321:
5317:
5313:
5309:
5305:
5301:
5297:
5293:
5289:
5288:
5287:
5285:
5281:
5277:
5267:
5265:
5260:
5258:
5254:
5250:
5246:
5242:
5220:
5217:
5214:
5210:
5205:
5202:
5199:
5193:
5190:
5187:
5182:
5179:
5176:
5173:
5168:
5164:
5157:
5151:
5145:
5138:
5137:
5136:
5134:
5115:
5112:
5109:
5106:
5103:
5100:
5097:
5094:
5091:
5088:
5085:
5082:
5079:
5059:
5056:
5053:
5030:
5027:
5024:
5017:
5010:
5007:
5004:
5001:
4996:
4992:
4985:
4979:
4973:
4964:
4935:
4932:
4929:
4920:
4917:
4914:
4908:
4900:
4897:
4894:
4885:
4882:
4879:
4870:
4864:
4861:
4858:
4853:
4849:
4845:
4842:
4837:
4833:
4827:
4824:
4821:
4818:
4815:
4810:
4806:
4799:
4793:
4787:
4780:
4779:
4778:
4776:
4772:
4768:
4757:
4753:
4736:
4733:
4728:
4724:
4720:
4713:
4709:
4705:
4699:
4693:
4687:
4680:
4677:
4674:
4673:
4658:
4655:
4652:
4649:
4646:
4643:
4636:
4620:
4617:
4612:
4609:
4606:
4603:
4600:
4597:
4592:
4588:
4585:
4582:
4579:
4576:
4571:
4567:
4563:
4557:
4551:
4545:
4538:
4535:
4531:
4528:
4525:
4524:
4507:
4504:
4499:
4496:
4489:
4472:
4469:
4466:
4463:
4458:
4454:
4450:
4445:
4442:
4437:
4433:
4429:
4423:
4417:
4411:
4404:
4401:
4398:
4395:
4394:
4379:
4376:
4373:
4366:
4349:
4346:
4341:
4337:
4332:
4327:
4321:
4315:
4308:
4294:
4291:
4288:
4281:
4278:
4277:
4273:
4270:
4267:
4264:
4263:
4257:
4254:
4249:
4247:
4237:
4236:tends to +∞.
4235:
4230:
4226:
4205:
4202:
4199:
4191:
4185:
4177:
4171:
4168:
4165:
4159:
4153:
4142:
4136:
4128:
4125:
4118:
4117:
4101:
4098:
4093:
4089:
4086:
4083:
4072:
4066:
4058:
4055:
4051:
4044:
4038:
4030:
4024:
4016:
4013:
4006:
4005:
4004:
4001:
3998:) = ln
3997:
3993:
3989:The function
3987:
3986:tends to +∞.
3985:
3981:
3977:
3971:
3967:
3947:
3944:
3940:
3936:
3933:
3930:
3925:
3921:
3918:
3915:
3912:
3909:
3904:
3900:
3896:
3889:
3880:
3874:
3866:
3860:
3857:
3854:
3848:
3842:
3831:
3825:
3817:
3814:
3807:
3792:
3789:
3782:
3778:
3773:
3770:
3767:
3764:
3761:
3756:
3752:
3748:
3737:
3731:
3723:
3720:
3716:
3709:
3703:
3695:
3689:
3681:
3678:
3671:
3670:
3669:
3666:
3662:
3658:
3654:
3650:
3644:
3642:
3638:
3634:
3630:
3625:
3621:
3617:
3612:
3608:
3586:
3583:
3580:
3574:
3568:
3560:
3554:
3536:
3528:
3521:
3520:
3519:
3517:
3513:
3508:
3491:
3468:
3460:
3441:
3437:
3430:
3424:
3419:
3413:
3395:
3387:
3380:
3379:
3378:
3376:
3372:
3368:
3364:
3360:
3356:
3346:
3324:
3321:
3316:
3313:
3303:
3297:
3289:
3282:
3267:
3263:
3260:
3256:
3250:
3247:
3242:
3239:
3235:
3230:
3221:
3215:
3207:
3200:
3185:
3181:
3178:
3172:
3166:
3162:
3153:
3147:
3135:
3134:
3133:
3131:
3127:
3123:
3120: =
3119:
3115:
3111:
3107:
3103:
3098:
3096:
3092:
3088:
3083:
3079:
3075:
3070:
3066:
3062:
3057:
3053:
3049:
3029:
3026:
3022:
3015:
3012:
3009:
3006:
3000:
2994:
2988:
2984:
2975:
2969:
2953:
2950:
2946:
2939:
2936:
2933:
2930:
2924:
2918:
2912:
2908:
2899:
2893:
2881:
2880:
2879:
2877:
2872:
2868:
2864:
2859:
2855:
2852:. A function
2851:
2847:
2843:
2839:
2831:
2827:
2823:
2819:
2800:
2797:
2791:
2788:
2785:
2779:
2773:
2764:
2755:
2753:
2749:
2733:
2725:
2721:
2717:
2711:
2708:
2702:
2695:
2691:
2688:(that has an
2686:
2682:
2662:
2659:
2653:
2650:
2645:
2641:
2636:
2626:
2620:
2612:
2606:
2603:
2598:
2594:
2589:
2579:
2573:
2561:
2560:
2559:
2549:
2545:
2539:
2535:
2528:
2522:
2512:
2503:
2493:
2473:
2468:
2465:
2460:
2454:
2448:
2445:
2437:
2431:
2404:
2401:
2396:
2393:
2387:
2381:
2378:
2370:
2364:
2352:
2351:
2350:
2348:
2343:
2337:
2333:
2330: =
2329:
2325:
2321:
2313:
2309:
2306: =
2305:
2301:
2297:
2277:
2274:
2268:
2262:
2254:
2248:
2223:
2220:
2214:
2208:
2200:
2194:
2182:
2181:
2180:
2178:
2174:
2171: =
2170:
2166:
2163: =
2162:
2156:
2151:
2144:
2139:
2130:
2116:
2113:
2110:
2087:
2080:
2077:
2053:
2030:
2010:
1989:
1986:
1977:
1961:
1936:
1932:
1924:
1921:
1900:
1897:
1894:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1825:
1819:
1813:
1810:
1807:
1804:
1794:
1786:
1783:
1774:
1769:
1765:
1757:
1756:
1755:
1733:
1729:
1721:
1718:
1709:
1706:
1697:
1694:
1688:
1682:
1676:
1669:
1666:
1658:
1657:
1656:
1642:
1615:
1612:
1608:
1603:
1597:
1594:
1588:
1584:
1581:
1578:
1572:
1569:
1562:
1554:
1550:
1543:
1535:
1529:
1523:
1516:
1512:
1505:
1493:
1492:
1491:
1474:
1468:
1465:
1461:
1456:
1450:
1447:
1441:
1437:
1434:
1431:
1425:
1422:
1415:
1407:
1403:
1396:
1388:
1382:
1376:
1369:
1365:
1358:
1346:
1345:
1344:
1330:
1327:
1324:
1321:
1297:
1294:
1291:
1263:
1260:
1251:
1248:
1245:
1239:
1236:
1230:
1224:
1218:
1211:
1210:
1209:
1206:
1203:
1201:
1197:
1193:
1187:
1182:
1178:
1172:
1145:
1142:
1139:
1129:
1122:
1119:
1116:
1113:
1101:
1098:
1090:
1085:
1079:
1073:
1066:
1065:
1064:
1062:
1058:
1054:
1050:
1046:
1043:The function
1041:
1039:
1017:
1014:
1008:
1005:
1002:
998:
989:
985:
978:
966:
949:
946:
940:
937:
934:
930:
921:
917:
910:
898:
897:
896:
894:
890:
886:
882:
877:
875:
871:
851:
847:
840:
827:
823:
803:
799:
792:
764:
758:
755:
749:
743:
736:
732:
725:
713:
699:
693:
690:
684:
678:
671:
667:
660:
648:
647:
646:
642:
638:
634:
629:
625:
621:
611:
609:
605:
601:
597:
593:
589:
585:
581:
575:
571:
567:
562:
552:
550:
546:
542:
538:
522:
502:
494:
476:
473:
451:
431:
411:
390:
384:
381:
376:
373:
369:
346:
343:
338:
332:
326:
317:
315:
306:
302:
298:
279:
276:
270:
264:
258:
250:
241:
239:
235:
230:
228:
224:
220:
215:
210:
202:
196:
192:
188:
184:
180:
176:
172:
168:
163:
161:
157:
153:
149:
145:
140:
138:
134:
130:
126:
122:
118:
114:
108:
72:
68:
59:
52:
48:
44:
40:
35:
29:
22:
8453:
8447:
8438:
8432:
8405:
8399:
8371:, New York:
8368:
8362:
8343:
8338:
8330:
8325:
8313:
8308:
8297:
8292:
8284:
8279:
8251:
8244:
8227:
8221:
8211:
8203:
8178:
8167:
8159:
8158:D.E. Smith,
8154:
8146:
8142:
8108:
8104:
8098:
8090:
8087:"Asymptotes"
8080:
8058:
8023:
7760:
7757:
7649:
7647:
7537:
7457:
7409:
7326:
7308:
7304:
7240:
7236:
7229:
7225:
7220:
7216:
7214:
7086:
6858:
6726:
6722:
6716:
6712:
6708:
6704:
6700:
6696:
6691:
6687:
6680:
6676:
6672:
6666:
6662:
6658:
6649:
6645:
6643:
6492:
6488:
6484:
6480:
6476:
6474:
6461:
6457:
6455:
6435:affine plane
6428:
6396:
6382:
6376:
6372:
6368:
6365:
6286:
6281:
6277:
6273:
6269:
6265:
6261:
6257:
6253:
6249:
6245:
6241:
6237:
6233:
6229:
6225:
6221:
6217:
6212:
6208:
6204:
6200:
6197:
6192:
6188:
6184:
6180:
6176:
6172:
6136:
6131:
6127:
6123:
6118:
6114:
6110:
6108:
6103:
6099:
6095:
6091:
6089:
6031:
6027:
6023:
6019:
6015:
6011:
6001:
5998:
5875:
5873:
5768:
5764:
5760:
5756:
5752:
5748:
5702:
5697:
5693:
5689:
5685:
5681:
5677:
5673:
5669:
5665:
5661:
5657:
5653:
5649:
5645:
5641:
5637:
5633:
5629:
5625:
5623:
5618:
5614:
5610:
5606:
5602:
5600:
5503:
5499:
5495:
5491:
5487:
5483:
5474:
5470:
5466:
5462:
5459:
5441:
5437:
5433:
5429:
5427:
5421:
5417:
5413:
5409:
5405:
5401:
5397:
5393:
5389:
5385:
5375:
5370:
5366:
5362:
5358:
5354:
5350:
5346:
5342:
5338:
5334:
5327:
5323:
5319:
5315:
5311:
5307:
5303:
5299:
5295:
5291:
5283:
5279:
5275:
5273:
5263:
5261:
5256:
5252:
5248:
5244:
5240:
5238:
5130:
4774:
4770:
4766:
4763:
4529:
4399:
4250:
4243:
4233:
4228:
4224:
4221:
3999:
3995:
3991:
3988:
3983:
3979:
3975:
3969:
3965:
3962:
3664:
3660:
3656:
3652:
3648:
3645:
3640:
3636:
3632:
3628:
3623:
3619:
3615:
3610:
3606:
3604:
3515:
3511:
3509:
3458:
3456:
3374:
3370:
3366:
3362:
3358:
3354:
3352:
3344:
3129:
3125:
3121:
3117:
3113:
3109:
3105:
3101:
3099:
3094:
3090:
3086:
3081:
3077:
3073:
3068:
3064:
3060:
3055:
3051:
3047:
3044:
2875:
2870:
2866:
2862:
2857:
2853:
2849:
2845:
2841:
2837:
2835:
2829:
2825:
2821:
2817:
2684:
2680:
2677:
2547:
2543:
2537:
2533:
2529:
2520:
2510:
2501:
2491:
2489:So the line
2488:
2344:
2335:
2331:
2327:
2323:
2319:
2311:
2307:
2303:
2299:
2295:
2293:
2176:
2172:
2168:
2164:
2160:
2154:
2149:
2148:
1886:
1753:
1634:
1489:
1313:
1207:
1204:
1191:
1185:
1180:
1176:
1170:
1167:
1060:
1056:
1052:
1048:
1044:
1042:
1037:
1035:
892:
888:
884:
880:
878:
873:
869:
825:
821:
779:
640:
636:
632:
627:
623:
619:
617:
607:
603:
599:
595:
591:
587:
583:
573:
569:
565:
558:
544:
540:
536:
492:
318:
310:
304:
300:
244:Introduction
234:in the large
233:
231:
226:
222:
218:
216:
208:
200:
194:
190:
186:
174:
170:
166:
164:
147:
146:ἀσύμπτωτος (
141:
123:coordinates
120:
116:
70:
64:
50:
46:
42:
38:
28:Asymptomatic
8209:"Asymptote"
8055:"Asymptote"
7540:hyperboloid
6655:homogeneous
6419:cubic curve
6106:can be +∞.
5664:-axis is 1/
5243:increases,
4227:= ln
872:approaches
295:graphed on
8502:Categories
8476:PlanetMath
8329:Frost, P.
8296:Frost, P.
8274:, p. 89ff.
8042:References
8018:defines a
7905:of degree
7874:where the
6657:of degree
6441:through a
6211:) →
5480:parametric
5473:) →
3461:is either
2750:, and the
2347:arctangent
2143:arctangent
1200:continuous
584:horizontal
167:horizontal
148:asumptōtos
8491:from the
8472:Asymptote
8452:P. Frost
8427:, p. 121.
8113:CiteSeerX
8065:EMS Press
7941:−
7843:⋯
7811:−
7713:−
7686:−
7603:−
7576:−
7493:−
7427:±
7365:−
7329:hyperbola
7288:≤
7266:≤
7171:−
7070:≠
7047:−
6818:−
6588:⋯
6562:−
6203: : (
6098:), where
6048:↦
5933:γ
5906:γ
5688:-axis is
5583:∞
5525:→
5465: : (
4933:−
4918:−
4898:−
4883:−
4843:−
4816:−
4769:= 0, and
4203:
4195:∞
4189:→
4166:−
4146:∞
4140:→
4087:
4076:∞
4070:→
4034:∞
4028:→
3931:−
3884:∞
3878:→
3855:−
3835:∞
3829:→
3741:∞
3735:→
3699:∞
3693:→
3639:tends to
3581:−
3558:→
3495:∞
3472:∞
3469:−
3417:→
3307:∞
3304:±
3301:→
3261:−
3225:∞
3222:±
3219:→
3179:−
3157:∞
3154:±
3151:→
3124:(that is
3001:−
2979:∞
2976:−
2973:→
2925:−
2903:∞
2897:→
2718:−
2712:
2706:↦
2690:hyperbola
2630:∞
2624:→
2583:∞
2580:−
2577:→
2523:tends to
2504:tends to
2466:π
2449:
2441:∞
2435:→
2402:π
2397:−
2382:
2374:∞
2371:−
2368:→
2338:tends to
2314:tends to
2258:∞
2252:→
2204:∞
2201:−
2198:→
2057:∞
2054:−
2034:∞
1872:…
1820:π
1784:−
1698:
1689:−
1619:∞
1616:−
1585:
1555:−
1547:→
1517:−
1509:→
1472:∞
1438:
1400:→
1362:→
1343:because
1252:
1173:→ 0
1143:≤
1021:∞
1018:−
1006:−
990:−
982:→
953:∞
938:−
914:→
844:→
804:−
796:→
762:∞
759:±
729:→
697:∞
694:±
672:−
664:→
618:The line
227:asymptote
211:tends to
203:tends to
71:asymptote
8485:Archived
8198:, §4.18.
8175:(1967),
8030:See also
7141:′
7110:′
6997:′
6966:′
6921:′
6876:′
6785:′
6751:′
6719:−1
6707:−
6387:parabola
6240:)), and
6135:, where
5404:), then
5349:), then
5306:), then
5133:dividing
4271:Example
4114:and then
3963:so that
3805:and then
2552:−∞
2316:−∞
2081:′
2069:. Hence
1990:′
1925:′
1670:′
1136:if
1110:if
588:vertical
561:calculus
183:function
171:vertical
8424:2156630
8391:0120551
8300:(1918)
8236:0114163
8135:2690881
7243:- 1 = 0
6433:in the
5648:(where
5378:scaling
5278:=0 for
4758:exists
4675:> 1
4279:< 0
3982:) when
3510:Having
3093:) when
3067:) when
2820:-axis (
2536:) = 1/(
592:oblique
175:oblique
133:tangent
111:) of a
8456:(1875)
8422:
8412:
8389:
8379:
8353:
8302:online
8267:
8234:
8191:
8133:
8115:
6644:where
4253:degree
3992:ƒ
3655:) = (2
3649:ƒ
3605:where
3457:where
2816:, the
2508:, and
2446:arctan
2379:arctan
2326:) has
2302:) has
780:where
637:ƒ
580:limits
570:ƒ
299:. The
191:ƒ
8239:, §8.
8131:JSTOR
6703:) = (
6685:, if
6470:conic
6451:field
6006:of a
6004:graph
5874:if γ(
5478:be a
4777:= 2.
4678:none
3663:+ 1)/
3635:) as
2840:- or
2179:) if
1040:= 1.
626:is a
549:limit
181:of a
179:graph
152:priv.
144:Greek
127:. In
113:curve
69:, an
8410:ISBN
8377:ISBN
8351:ISBN
8265:ISBN
8189:ISBN
8020:cone
7925:and
7901:are
7327:The
7311:=0.
7031:and
6904:and
6198:Let
6191:+4)/
6139:and
5460:Let
5282:(x)=
4526:= 1
4396:= 0
4251:The
4003:has
3668:has
3112:+ 1/
2746:the
2683:↦ 1/
2554:and
2419:and
2141:The
2046:nor
1839:for
1490:and
1117:>
883:) =
590:and
543:and
314:Line
303:and
173:and
8474:at
8257:hdl
8123:doi
6653:is
6379:+ 3
6375:+ 2
5518:lim
5410:cax
5384:If
5333:If
5290:If
4222:So
4182:lim
4133:lim
4063:lim
4021:lim
3972:+ 3
3968:= 2
3871:lim
3822:lim
3728:lim
3686:lim
3659:+ 3
3551:lim
3542:def
3484:or
3410:lim
3401:def
3294:lim
3212:lim
3144:lim
2966:lim
2890:lim
2848:or
2709:exp
2617:lim
2570:lim
2540:+1)
2494:= –
2428:lim
2361:lim
2245:lim
2236:or
2191:lim
2003:at
1978:of
1695:cos
1582:sin
1540:lim
1502:lim
1435:sin
1393:lim
1355:lim
1283:at
1249:sin
1188:= 0
975:lim
907:lim
837:lim
789:lim
722:lim
657:lim
119:or
65:In
8504::
8420:MR
8418:,
8387:MR
8385:,
8375:,
8263:,
8232:MR
8187:,
8129:,
8121:,
8109:72
8107:,
8089:,
8063:,
8057:,
7743:0.
7523:0.
7239:+
6725:,
6711:)
6709:by
6705:ax
6699:,
6675:=
6453:.
6421:,
6417:A
6371:=
6232:),
6187:+3
6183:+2
6175:+2
6130:+
6128:mx
6126:=
5763:),
5640:,
5498:),
5436:)=
5418:cf
5414:cb
5390:ax
5369:)+
5251:=
4473:12
4244:A
4200:ln
4084:ln
3643:.
3622:+
3620:mx
3618:=
3369:+
3367:mx
3365:=
3325:0.
3080:+
3078:mx
3076:=
3054:+
3052:mx
3050:=
3030:0.
2869:+
2867:mx
2865:=
2828:=
2754:.
2663:0.
2556:+∞
2532:ƒ(
2527:.
2525:+∞
2517:/2
2513:=
2506:–∞
2498:/2
2342:.
2340:+∞
2129:.
1175:,
1146:0.
1059:=
887:/(
635:=
622:=
586:,
568:=
240:.
189:=
169:,
139:.
101:oʊ
53:).
8478:.
8357:,
8320:)
8259::
8125::
8006:0
8003:=
8000:)
7997:z
7994:,
7991:y
7988:,
7985:x
7982:(
7977:d
7973:P
7952:0
7949:=
7944:1
7938:d
7934:P
7913:i
7887:i
7883:P
7862:,
7859:0
7856:=
7851:0
7847:P
7840:+
7837:)
7834:z
7831:,
7828:y
7825:,
7822:x
7819:(
7814:2
7808:d
7804:P
7800:+
7797:)
7794:z
7791:,
7788:y
7785:,
7782:x
7779:(
7774:d
7770:P
7740:=
7733:2
7729:c
7723:2
7719:z
7706:2
7702:b
7696:2
7692:y
7679:2
7675:a
7669:2
7665:x
7633:1
7630:=
7623:2
7619:c
7613:2
7609:z
7596:2
7592:b
7586:2
7582:y
7569:2
7565:a
7559:2
7555:x
7520:=
7513:2
7509:b
7503:2
7499:y
7486:2
7482:a
7476:2
7472:x
7443:.
7440:x
7435:a
7432:b
7424:=
7421:y
7395:1
7392:=
7385:2
7381:b
7375:2
7371:y
7358:2
7354:a
7348:2
7344:x
7309:x
7305:x
7291:1
7284:|
7280:y
7276:|
7272:,
7269:1
7262:|
7258:x
7254:|
7241:y
7237:x
7230:n
7226:P
7221:n
7217:P
7200:,
7197:0
7194:=
7191:)
7188:a
7185:,
7182:b
7179:(
7174:1
7168:n
7164:P
7160:=
7157:)
7154:a
7151:,
7148:b
7145:(
7137:y
7133:Q
7129:=
7126:)
7123:a
7120:,
7117:b
7114:(
7106:x
7102:Q
7073:0
7067:)
7064:a
7061:,
7058:b
7055:(
7050:1
7044:n
7040:P
7019:0
7016:=
7013:)
7010:a
7007:,
7004:b
7001:(
6993:y
6989:Q
6985:=
6982:)
6979:a
6976:,
6973:b
6970:(
6962:x
6958:Q
6937:)
6934:a
6931:,
6928:b
6925:(
6917:y
6913:Q
6892:)
6889:a
6886:,
6883:b
6880:(
6872:x
6868:Q
6844:0
6841:=
6838:)
6835:a
6832:,
6829:b
6826:(
6821:1
6815:n
6811:P
6807:+
6804:y
6801:)
6798:a
6795:,
6792:b
6789:(
6781:y
6777:Q
6773:+
6770:x
6767:)
6764:a
6761:,
6758:b
6755:(
6747:x
6743:Q
6729:)
6727:y
6723:x
6721:(
6717:n
6713:Q
6701:y
6697:x
6695:(
6692:n
6688:P
6681:n
6677:P
6673:Q
6667:n
6663:P
6659:k
6650:k
6646:P
6627:0
6623:P
6619:+
6616:)
6613:y
6610:,
6607:x
6604:(
6599:1
6595:P
6591:+
6585:+
6582:)
6579:y
6576:,
6573:x
6570:(
6565:1
6559:n
6555:P
6551:+
6548:)
6545:y
6542:,
6539:x
6536:(
6531:n
6527:P
6523:=
6520:)
6517:y
6514:,
6511:x
6508:(
6505:P
6493:n
6489:P
6485:y
6483:,
6481:x
6479:(
6477:P
6462:n
6458:n
6377:x
6373:x
6369:y
6349:x
6345:4
6342:+
6339:x
6336:3
6333:+
6328:2
6324:x
6320:2
6317:+
6312:3
6308:x
6301:=
6298:y
6282:A
6278:B
6274:b
6270:t
6266:B
6262:t
6260:(
6258:A
6254:A
6250:B
6246:A
6242:B
6238:t
6236:(
6234:y
6230:t
6228:(
6226:x
6222:t
6220:(
6218:A
6213:R
6209:b
6207:,
6205:a
6201:A
6193:x
6189:x
6185:x
6181:x
6177:x
6173:x
6147:n
6137:m
6132:n
6124:y
6119:c
6115:c
6111:x
6104:b
6100:a
6096:b
6094:,
6092:a
6075:.
6072:)
6069:)
6066:t
6063:(
6060:f
6057:,
6054:t
6051:(
6045:t
6032:x
6030:(
6028:ƒ
6026:,
6024:x
6020:x
6018:(
6016:ƒ
6012:y
5979:2
5975:b
5971:+
5966:2
5962:a
5955:|
5951:c
5948:+
5945:)
5942:)
5939:t
5936:(
5930:(
5927:y
5924:b
5921:+
5918:)
5915:)
5912:t
5909:(
5903:(
5900:x
5897:a
5893:|
5876:t
5854:2
5850:b
5846:+
5841:2
5837:a
5830:|
5826:c
5823:+
5820:)
5817:t
5814:(
5811:y
5808:b
5805:+
5802:)
5799:t
5796:(
5793:x
5790:a
5786:|
5769:t
5767:(
5765:y
5761:t
5759:(
5757:x
5753:t
5751:(
5749:A
5735:0
5732:=
5729:c
5726:+
5723:y
5720:b
5717:+
5714:x
5711:a
5698:y
5694:t
5690:t
5686:y
5682:t
5678:y
5674:x
5670:t
5666:t
5662:x
5658:t
5654:x
5650:t
5646:t
5642:y
5638:t
5634:x
5630:x
5626:y
5619:b
5615:t
5611:t
5609:(
5607:A
5603:A
5586:.
5580:=
5577:)
5574:)
5571:t
5568:(
5563:2
5559:y
5555:+
5552:)
5549:t
5546:(
5541:2
5537:x
5533:(
5528:b
5522:t
5504:t
5502:(
5500:y
5496:t
5494:(
5492:x
5488:t
5486:(
5484:A
5475:R
5471:b
5469:,
5467:a
5463:A
5442:y
5438:e
5434:x
5432:(
5430:f
5424:)
5422:x
5420:(
5412:+
5408:=
5406:y
5402:x
5400:(
5398:f
5394:b
5392:+
5388:=
5386:y
5371:k
5367:x
5365:(
5363:f
5359:k
5357:+
5355:c
5353:=
5351:y
5347:x
5345:(
5343:f
5339:c
5337:=
5335:y
5330:)
5328:h
5326:-
5324:x
5322:(
5320:f
5316:h
5314:+
5312:a
5310:=
5308:x
5304:x
5302:(
5300:f
5296:a
5294:=
5292:x
5284:e
5280:f
5276:y
5264:x
5257:x
5253:x
5249:y
5245:f
5241:x
5221:1
5218:+
5215:x
5211:1
5206:+
5203:x
5200:=
5194:1
5191:+
5188:x
5183:1
5180:+
5177:x
5174:+
5169:2
5165:x
5158:=
5155:)
5152:x
5149:(
5146:f
5116:6
5113:,
5110:5
5107:,
5104:4
5101:,
5098:3
5095:,
5092:2
5089:,
5086:1
5083:=
5080:x
5060:x
5057:=
5054:y
5034:)
5031:1
5028:+
5025:x
5022:(
5018:/
5014:)
5011:1
5008:+
5005:x
5002:+
4997:2
4993:x
4989:(
4986:=
4983:)
4980:x
4977:(
4974:f
4939:)
4936:2
4930:x
4927:(
4924:)
4921:1
4915:x
4912:(
4909:x
4904:)
4901:3
4895:x
4892:(
4889:)
4886:2
4880:x
4877:(
4871:=
4865:x
4862:2
4859:+
4854:2
4850:x
4846:3
4838:3
4834:x
4828:6
4825:+
4822:x
4819:5
4811:2
4807:x
4800:=
4797:)
4794:x
4791:(
4788:f
4775:x
4771:x
4767:x
4737:1
4734:+
4729:2
4725:x
4721:3
4714:4
4710:x
4706:2
4700:=
4697:)
4694:x
4691:(
4688:f
4659:3
4656:+
4653:x
4650:2
4647:=
4644:y
4621:x
4618:5
4613:+
4610:3
4607:+
4604:x
4601:2
4598:=
4593:x
4589:5
4586:+
4583:x
4580:3
4577:+
4572:2
4568:x
4564:2
4558:=
4555:)
4552:x
4549:(
4546:f
4530:y
4508:3
4505:2
4500:=
4497:y
4470:+
4467:x
4464:+
4459:2
4455:x
4451:3
4446:7
4443:+
4438:2
4434:x
4430:2
4424:=
4421:)
4418:x
4415:(
4412:f
4400:y
4380:0
4377:=
4374:y
4350:1
4347:+
4342:2
4338:x
4333:1
4328:=
4325:)
4322:x
4319:(
4316:f
4295:0
4292:=
4289:y
4234:x
4229:x
4225:y
4206:x
4192:+
4186:x
4178:=
4175:)
4172:x
4169:m
4163:)
4160:x
4157:(
4154:f
4151:(
4143:+
4137:x
4129:=
4126:n
4102:0
4099:=
4094:x
4090:x
4073:+
4067:x
4059:=
4056:x
4052:/
4048:)
4045:x
4042:(
4039:f
4031:+
4025:x
4017:=
4014:m
4000:x
3996:x
3994:(
3984:x
3980:x
3978:(
3976:ƒ
3970:x
3966:y
3948:3
3945:=
3941:)
3937:x
3934:2
3926:x
3922:1
3919:+
3916:x
3913:3
3910:+
3905:2
3901:x
3897:2
3890:(
3881:+
3875:x
3867:=
3864:)
3861:x
3858:m
3852:)
3849:x
3846:(
3843:f
3840:(
3832:+
3826:x
3818:=
3815:n
3793:2
3790:=
3783:2
3779:x
3774:1
3771:+
3768:x
3765:3
3762:+
3757:2
3753:x
3749:2
3738:+
3732:x
3724:=
3721:x
3717:/
3713:)
3710:x
3707:(
3704:f
3696:+
3690:x
3682:=
3679:m
3665:x
3661:x
3657:x
3653:x
3651:(
3641:a
3637:x
3633:x
3631:(
3629:ƒ
3624:n
3616:y
3611:m
3607:a
3590:)
3587:x
3584:m
3578:)
3575:x
3572:(
3569:f
3566:(
3561:a
3555:x
3537:=
3529:n
3516:n
3512:m
3492:+
3459:a
3442:x
3438:/
3434:)
3431:x
3428:(
3425:f
3420:a
3414:x
3396:=
3388:m
3375:m
3371:n
3363:y
3359:x
3357:(
3355:f
3322:=
3317:x
3314:1
3298:x
3290:=
3268:]
3264:x
3257:)
3251:x
3248:1
3243:+
3240:x
3236:(
3231:[
3216:x
3208:=
3186:]
3182:x
3176:)
3173:x
3170:(
3167:f
3163:[
3148:x
3130:n
3126:m
3122:x
3118:y
3114:x
3110:x
3106:x
3104:(
3102:ƒ
3095:x
3091:x
3089:(
3087:ƒ
3082:n
3074:y
3069:x
3065:x
3063:(
3061:ƒ
3056:n
3048:y
3027:=
3023:]
3019:)
3016:n
3013:+
3010:x
3007:m
3004:(
2998:)
2995:x
2992:(
2989:f
2985:[
2970:x
2954:0
2951:=
2947:]
2943:)
2940:n
2937:+
2934:x
2931:m
2928:(
2922:)
2919:x
2916:(
2913:f
2909:[
2900:+
2894:x
2876:m
2874:(
2871:n
2863:y
2858:x
2856:(
2854:ƒ
2842:y
2838:x
2830:x
2826:y
2822:x
2818:y
2801:x
2798:1
2792:+
2789:x
2786:=
2783:)
2780:x
2777:(
2774:f
2734:,
2731:)
2726:2
2722:x
2715:(
2703:x
2685:x
2681:x
2660:=
2654:1
2651:+
2646:2
2642:x
2637:1
2627:+
2621:x
2613:=
2607:1
2604:+
2599:2
2595:x
2590:1
2574:x
2548:x
2544:y
2538:x
2534:x
2521:x
2515:π
2511:y
2502:x
2496:π
2492:y
2474:.
2469:2
2461:=
2458:)
2455:x
2452:(
2438:+
2432:x
2405:2
2394:=
2391:)
2388:x
2385:(
2365:x
2336:x
2332:c
2328:y
2324:x
2322:(
2320:ƒ
2312:x
2308:c
2304:y
2300:x
2298:(
2296:ƒ
2290:.
2278:c
2275:=
2272:)
2269:x
2266:(
2263:f
2255:+
2249:x
2224:c
2221:=
2218:)
2215:x
2212:(
2209:f
2195:x
2177:x
2175:(
2173:ƒ
2169:y
2165:c
2161:y
2155:x
2117:0
2114:=
2111:x
2091:)
2088:x
2085:(
2078:f
2031:+
2011:0
1987:f
1962:0
1942:)
1937:n
1933:x
1929:(
1922:f
1901:0
1898:=
1895:x
1869:,
1866:2
1863:,
1860:1
1857:,
1854:0
1851:=
1848:n
1826:,
1817:)
1814:1
1811:+
1808:n
1805:2
1802:(
1795:n
1791:)
1787:1
1781:(
1775:=
1770:n
1766:x
1750:.
1734:2
1730:x
1725:)
1722:1
1719:+
1716:)
1710:x
1707:1
1701:(
1692:(
1683:=
1680:)
1677:x
1674:(
1667:f
1643:f
1631:.
1613:=
1609:)
1604:)
1598:x
1595:1
1589:(
1579:+
1573:x
1570:1
1563:(
1551:0
1544:x
1536:=
1533:)
1530:x
1527:(
1524:f
1513:0
1506:x
1475:,
1469:+
1466:=
1462:)
1457:)
1451:x
1448:1
1442:(
1432:+
1426:x
1423:1
1416:(
1408:+
1404:0
1397:x
1389:=
1386:)
1383:x
1380:(
1377:f
1370:+
1366:0
1359:x
1331:,
1328:0
1325:=
1322:x
1310:.
1298:0
1295:=
1292:x
1270:)
1264:x
1261:1
1255:(
1246:+
1240:x
1237:1
1231:=
1228:)
1225:x
1222:(
1219:f
1192:ƒ
1186:x
1181:x
1179:(
1177:ƒ
1171:x
1140:x
1130:5
1123:,
1120:0
1114:x
1102:x
1099:1
1091:{
1086:=
1083:)
1080:x
1077:(
1074:f
1061:a
1057:x
1053:a
1049:x
1047:(
1045:ƒ
1038:x
1015:=
1009:1
1003:x
999:x
986:1
979:x
950:+
947:=
941:1
935:x
931:x
922:+
918:1
911:x
893:x
889:x
885:x
881:x
874:a
870:x
852:+
848:a
841:x
826:a
822:x
800:a
793:x
765:,
756:=
753:)
750:x
747:(
744:f
737:+
733:a
726:x
700:,
691:=
688:)
685:x
682:(
679:f
668:a
661:x
643:)
641:x
639:(
633:y
624:a
620:x
608:x
604:x
600:x
596:x
576:)
574:x
572:(
566:y
545:y
541:x
537:y
523:y
503:x
493:x
477:x
474:1
452:x
432:y
412:x
391:)
385:x
382:1
377:,
374:x
370:(
347:x
344:1
339:=
336:)
333:x
330:(
327:f
305:y
301:x
280:x
277:1
271:=
268:)
265:x
262:(
259:f
209:x
201:x
197:)
195:x
193:(
187:y
121:y
117:x
107:/
104:t
98:t
95:p
92:m
89:ɪ
86:s
83:æ
80:ˈ
77:/
73:(
51:x
47:y
43:x
39:y
30:.
23:.
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