return - Maple Help

limit/return

values returned by limit

 Calling Sequence limit(f, point, dir, parametric)

Parameters

 f - algebraic expression point - equation, $x=a$, where x is a name and a is the limit point dir - (optional) direction; either left, right, real (default) or complex parametric - (optional) literal name

Description

 • The meaning of a returned $\mathrm{\infty }$ depends on dir. If dir is complex, then $\mathrm{\infty }$ denotes complex infinity.  Otherwise, the result $\mathrm{\infty }$ denotes real positive infinity, and $-\mathrm{\infty }$ denotes real negative infinity.
 • A return value of $\mathrm{\infty }$ means that when $x$ approaches $a$, then the real part of $f$ approaches $\mathrm{\infty }$, but the imaginary part of $f$ remains bounded, and similarly for other improper limits such as $-\mathrm{\infty }$, $I\mathrm{\infty }$, and $-I\mathrm{\infty }$. If specifically the limit of the real or imaginary part of $f$ is desired, compute the limit of $\mathrm{\Re }\left(f\right)$ or $\mathrm{\Im }\left(f\right)$, respectively, instead.
 • A return value of $c\mathrm{\infty }$ for a finite complex constant $c$ means that $f$ approaches the North pole of the Riemann sphere along the ray ${ℝ}_{\ge 0}$.
 • If limit returns a numeric range it means that the value of the limiting expression is known to lie in that range for arguments restricted to some neighborhood of the limit point.  It does not necessarily imply that the limiting expression is known to achieve every value infinitely often in this range.
 • If the limit is known to be undefined, or each side of a two-sided limit has a different value, or for a multi-dimensional limit the limiting value depends on the direction from which the limit is approached, then undefined is returned.
 • If limit is unable to evaluate the limit, then it returns unevaluated. Unless the limit is unevaluated, the limit returned is independent of the variable given in point.
 • When option parametric is specified, often a piecewise expression is returned. One of the branches in this piecewise expression (typically the last one, or otherwise branch) may be an inert Limit.

Examples

 > $\mathrm{limit}\left(\mathrm{exp}\left(x\right),x=\mathrm{\infty }\right)$
 ${\mathrm{\infty }}$ (1)
 > $\mathrm{limit}\left(\frac{1}{x},x=0,\mathrm{complex}\right)$
 ${\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}$ (2)
 > $\mathrm{limit}\left(\mathrm{ln}\left(x\right),x=-\mathrm{\infty }\right)$
 ${\mathrm{\infty }}$ (3)
 > $\mathrm{limit}\left(\mathrm{\Im }\left(\mathrm{ln}\left(x\right)\right),x=-\mathrm{\infty }\right)$
 ${\mathrm{\pi }}$ (4)
 > $\mathrm{limit}\left(\mathrm{exp}\left(x+\frac{I\mathrm{\pi }}{3}+\frac{1}{x}\right),x=\mathrm{\infty }\right)$
 $\left({1}{+}{I}{}\sqrt{{3}}\right){}{\mathrm{\infty }}$ (5)
 > $\mathrm{limit}\left(\mathrm{sin}\left(\frac{1}{x}\right),x=0\right)$
 ${-1}{..}{1}$ (6)
 > $\mathrm{limit}\left(\mathrm{exp}\left(x\right),x=\mathrm{\infty },\mathrm{real}\right)$
 ${\mathrm{undefined}}$ (7)
 > $\mathrm{limit}\left(\mathrm{tan}\left(x\right),x=\mathrm{\infty }\right)$
 ${\mathrm{undefined}}$ (8)
 > $\mathrm{limit}\left(ax,x=\mathrm{\infty }\right)$
 ${\mathrm{signum}}{}\left({a}\right){}{\mathrm{\infty }}$ (9)
 > $\mathrm{limit}\left({x}^{a},x=\mathrm{\infty },'\mathrm{parametric}'\right)$
 $\left\{\begin{array}{cc}{0}& {a}{<}{0}\\ {1}& {a}{=}{0}\\ {\mathrm{\infty }}& {0}{<}{a}\\ \underset{{x}{\to }{\mathrm{\infty }}}{{lim}}{}{{x}}^{{a}}& {\mathrm{\Im }}{}\left({a}\right){\ne }{0}\end{array}\right\$ (10)