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limit/multi

multidimensional limits

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

Parameters

 f - algebraic expression point - set or list of equations of the form $x=a$ dir - (optional) direction

Description

 • The limit(f, point) calling sequence attempts to compute the limiting value of f in a multidimensional space at point.
 • An equation of the form $x=x$ in the limiting point is ignored.
 • If the limit depends on the direction of approach, either a range or undefined is returned (see limit,return).
 • The optional direction dir is applied to all coordinates in point, and consequently each coordinate is approached from the same direction; from the left or right, bidirectional (real), or complex.

Examples

 > $\mathrm{limit}\left(\frac{{x}^{2}-{y}^{2}}{{x}^{2}+{y}^{2}},\left\{x=0\right\}\right)$
 ${-1}$ (1)
 > $\mathrm{limit}\left(\frac{{x}^{2}-{y}^{2}}{{x}^{2}+{y}^{2}},\left[x=0,y=0\right]\right)$
 ${-1}{..}{1}$ (2)
 > $\mathrm{limit}\left(x+\frac{1}{y},\left\{x=0,y=\mathrm{\infty }\right\}\right)$
 ${0}$ (3)
 > $\mathrm{limit}\left(xy,\left\{x=0,y=\mathrm{\infty }\right\}\right)$
 ${\mathrm{undefined}}$ (4)
 > $\mathrm{limit}\left(\frac{{x}^{3}+{y}^{3}}{{x}^{2}+xy+{y}^{2}},\left\{x=0,y=0\right\}\right)$
 ${0}$ (5)
 > $\mathrm{limit}\left(\frac{{x}^{3}y}{2{x}^{4}+{y}^{4}},\left\{x=0,y=0\right\}\right)$
 ${-}\frac{{{6}}^{{3}}{{4}}}{}\sqrt{{2}}}{{16}}{..}\frac{{{6}}^{{3}}{{4}}}{}\sqrt{{2}}}{{16}}$ (6)
 > $\mathrm{limit}\left(\frac{{x}^{4}-{y}^{2}+3{x}^{2}y-{x}^{2}}{{x}^{2}+{y}^{2}},\left\{x=0,y=0\right\}\right)$
 ${-1}$ (7)
 > $\mathrm{limit}\left(\frac{\mathrm{sin}\left({x}^{2}\right)-\mathrm{sin}\left({y}^{2}\right)}{x-y},\left\{x=0,y=0\right\}\right)$
 ${0}$ (8)
 > $\mathrm{limit}\left(\frac{1}{y-x},\left\{x=y\right\}\right)$
 ${\mathrm{undefined}}$ (9)
 > $\mathrm{limit}\left(\frac{1}{y-x},\left\{x=y\right\},\mathrm{right}\right)$
 ${-}{\mathrm{\infty }}$ (10)
 > $\mathrm{limit}\left(\frac{1}{y-x},\left\{x=y\right\},\mathrm{left}\right)$
 ${\mathrm{\infty }}$ (11)

References

 C. Cadavid, S. Molina, J.D. Velez, Limits of quotients of bivariate real analytic functions, Journal of Symbolic Computation, Volume 50, March 2013, 197-207, doi:10.1016/j.jsc.2012.07.004
 Gary R. Lawlor, l'Hôpital's Rule for Multivariable Functions, The American Mathematical Monthly, 127:8 (2020), 717-725, doi:10.1080/00029890.2020.1793635

Compatibility

 • The limit/multi command was updated in Maple 2021.