Limit - Maple Help

limit

calculate limit

Limit

inert form of limit

Calling Sequence

 limit(f, x=a) $\underset{x\to a}{lim}f$ Limit(f, x=a) $\underset{x\to a}{{lim}}f$ limit(f, x=a, dir, parametric=p) Limit(f, x=a, dir)

Parameters

 f - algebraic expression x - name a - algebraic expression; limit point, possibly infinity, or -infinity dir - (optional) symbol; direction chosen from: left, right, real, or complex p - (optional) either true or false (default)

Options

 • parametric=p
 If parametric=true, or just parametric, is specified, then limit tries to compute an answer that is correct for all real values of any parameter(s) appearing in a. Typically, the result is a piecewise expression.

Description

 • The limit(f, x=a, dir) function attempts to compute the limiting value of f as x approaches a.
 • You can enter the command limit using either the 1-D or 2-D calling sequence.
 • If dir is not specified, the limit is the real bidirectional limit, except in the case where the limit point is infinity or -infinity, in which case the limit is from the left to infinity and from the right to -infinity. For help with directional limits, see limit/dir.
 • The output from limit can be a range (meaning a bounded result) or an algebraic expression, possibly containing infinity. For further help with the return type, see limit/return.
 • To compute a limit in a multidimensional space, specify a set of points as the second argument. For more information, see limit/multi.
 • Most limits are resolved by computing series. By increasing the value of the global variable Order, the ability of limit to solve problems with significant cancellation improves.
 • If Maple cannot find a closed form for the limit, the function calling sequence is returned.
 Note: The limit function always assumes that the limit variable approaches the limit point along (one or more) continuous paths (e.g., along the real axis from the left or from the right). It is not currently possible to compute limits where the limit variable takes only discrete or integer values.
 Also, the limit function ignores any assumptions on the limit variable made via assume or assuming.
 • If $f$ is a function not known to Maple, the limit function assumes that $f$ is regular at a finite expansion point.
 • The capitalized function name Limit is the inert limit function, which returns unevaluated.  It appears gray so that it is easily distinguished from a returned limit calling sequence.
 Note: Since Limit does not try to evaluate or check the existence of the limit of the expression, it can lead to incorrect transformations. Therefore, the use of limit is more reliable. This is demonstrated by the last two examples.

Examples

The inert Limit function returns unevaluated.

 > $\mathrm{Limit}\left(\frac{1}{x},x=0\right)$
 $\underset{{x}{\to }{0}}{{lim}}{}\frac{{1}}{{x}}$ (1)
 > $\mathrm{Limit}\left(\mathrm{ln}\left({x}^{2}\right),x=-\mathrm{\infty }\right)$
 $\underset{{x}{\to }{-}{\mathrm{\infty }}}{{lim}}{}{\mathrm{ln}}{}\left({{x}}^{{2}}\right)$ (2)
 > $\mathrm{limit}\left(\frac{1}{x},x=3\right)$
 $\frac{{1}}{{3}}$ (3)
 > $\mathrm{limit}\left({x}^{2},x=\mathrm{\infty }\right)$
 ${\mathrm{\infty }}$ (4)
 > $\mathrm{limit}\left(\frac{\mathrm{sin}\left(x\right)}{x},x=0\right)$
 ${1}$ (5)
 > $\mathrm{limit}\left(\mathrm{exp}\left(x\right),x=\mathrm{\infty }\right)$
 ${\mathrm{\infty }}$ (6)
 > $\mathrm{limit}\left(\mathrm{exp}\left(x\right),x=-\mathrm{\infty }\right)$
 ${0}$ (7)
 > $\mathrm{limit}\left(\mathrm{exp}\left({x}^{2}\right)\left(1-\mathrm{erf}\left(x\right)\right),x=\mathrm{\infty }\right)$
 ${0}$ (8)
 > $\mathrm{limit}\left(\frac{1}{x},x=\mathrm{\infty }\right)$
 ${0}$ (9)

To use the real and complex arguments, the limit command must be written in 1-D or 2-D command form and not in mathematical notation. See Entering Commands in 2-D Math for more information.

 > $\mathrm{limit}\left(\frac{1}{x},x=0,\mathrm{real}\right)$
 ${\mathrm{undefined}}$ (10)
 > $\mathrm{limit}\left(\frac{1}{x},x=0,\mathrm{complex}\right)$
 ${\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}$ (11)

 > $\mathrm{limit}\left(\frac{1}{x},x=0,\mathrm{right}\right)$
 ${\mathrm{\infty }}$ (12)
 > $\mathrm{limit}\left(\frac{1}{x},x=0,\mathrm{left}\right)$
 ${-}{\mathrm{\infty }}$ (13)
 > $g≔\mathrm{piecewise}\left(x<3,{x}^{2}-6,3\le x,2x-1\right)$
 ${g}{≔}\left\{\begin{array}{cc}{{x}}^{{2}}{-}{6}& {x}{<}{3}\\ {2}{}{x}{-}{1}& {3}{\le }{x}\end{array}\right\$ (14)
 > $\mathrm{limit}\left(g,x=3\right)$
 ${\mathrm{undefined}}$ (15)
 > $\mathrm{limit}\left(g,x=3,\mathrm{right}\right)$
 ${5}$ (16)
 > $\mathrm{limit}\left(g,x=3,\mathrm{left}\right)$
 ${3}$ (17)

Unknown functions are assumed to be regular at a finite expansion point.

 > $\mathrm{limit}\left(f\left(x\right),x=a\right)$
 ${f}{}\left({a}\right)$ (18)

Parametric examples:

 > $\mathrm{limit}\left({x}^{a},x=\mathrm{\infty }\right)$
 $\underset{{x}{\to }{\mathrm{\infty }}}{{lim}}{}{{x}}^{{a}}$ (19)
 > $\mathrm{limit}\left({x}^{a},x=\mathrm{\infty },'\mathrm{parametric}'\right)$
 $\left\{\begin{array}{cc}{0}& {a}{<}{0}\\ {1}& {a}{=}{0}\\ {\mathrm{\infty }}& {0}{<}{a}\\ \underset{{x}{\to }{\mathrm{\infty }}}{{lim}}{}{{x}}^{{a}}& {\mathrm{otherwise}}\end{array}\right\$ (20)
 > $\mathrm{limit}\left({x}^{a},x=0,'\mathrm{parametric}'\right)$
 $\left\{\begin{array}{cc}{1}& {a}{=}{0}\\ {0}& {0}{<}{a}\\ \underset{{x}{\to }{0}}{{lim}}{}{{x}}^{{a}}& {\mathrm{otherwise}}\end{array}\right\$ (21)
 > $\mathrm{limit}\left({x}^{a},x=0,'\mathrm{right}','\mathrm{parametric}'\right)$
 $\left\{\begin{array}{cc}{\mathrm{\infty }}& {a}{<}{0}\\ {1}& {a}{=}{0}\\ {0}& {0}{<}{a}\\ \underset{{x}{\to }{0}{+}}{{lim}}{}{{x}}^{{a}}& {\mathrm{otherwise}}\end{array}\right\$ (22)
 > $\mathrm{limit}\left({x}^{a},x=0,'\mathrm{left}','\mathrm{parametric}'\right)$
 $\left\{\begin{array}{cc}{{ⅇ}}^{{I}{}{\mathrm{\pi }}{}{a}}{}{\mathrm{\infty }}& {a}{<}{0}\\ {1}& {a}{=}{0}\\ {0}& {0}{<}{a}\\ \underset{{x}{\to }{0}{-}}{{lim}}{}{{x}}^{{a}}& {\mathrm{otherwise}}\end{array}\right\$ (23)
 > $\mathrm{limit}\left(\mathrm{tan}\left(x+\frac{a\mathrm{\pi }}{2}\right),x=0,'\mathrm{right}'\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a::'\mathrm{integer}'$
 $\underset{{x}{\to }{0}{+}}{{lim}}{}{\mathrm{tan}}{}\left({x}{+}\frac{{\mathrm{\pi }}{}{a}}{{2}}\right)$ (24)
 > $\mathrm{limit}\left(\mathrm{tan}\left(x+\frac{a\mathrm{\pi }}{2}\right),x=0,'\mathrm{right}','\mathrm{parametric}'\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a::'\mathrm{integer}'$
 $\left\{\begin{array}{cc}{0}& {a}{::}{\mathrm{even}}\\ {-}{\mathrm{\infty }}& {a}{::}{\mathrm{odd}}\end{array}\right\$ (25)

The use of the inert Limit function can lead to errors.

 > $\mathrm{combine}\left(\left(\mathrm{Limit}\left(\frac{1}{x},x=0\right)\right)\left(\mathrm{Limit}\left(x,x=0\right)\right)\right)$
 $\underset{{x}{\to }{0}}{{lim}}{}{1}$ (26)
 > $\mathrm{combine}\left(\left(\mathrm{limit}\left(\frac{1}{x},x=0\right)\right)\left(\mathrm{limit}\left(x,x=0\right)\right)\right)$
 ${\mathrm{undefined}}$ (27)

References

 Geddes, K. O., and Gonnet, G. H. "A New Algorithm for Computing Symbolic Limits Using Hierarchical Series." In Proceedings of ISSAC '88, pp. 490-495. Edited by Patrizia M. Gianni. Berlin: Springer-Verlag, 1988.

Compatibility

 • The limit command was updated in Maple 2016; see Advanced Math.
 • The limit command was updated in Maple 2020.
 • The parametric option was introduced in Maple 2020.