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.

$\mathrm{limit}\left(\frac{x^2-y^2}{x^2+y^2}\,\left\{x=0\right\}\right)$

$\mathrm{-1}$

$\mathrm{limit}\left(\frac{x^2-y^2}{x^2+y^2}\,\left[x=0\,y=0\right]\right)$

$\mathrm{-1}..1$

$\mathrm{limit}\left(x+\frac{1}{y}\,\left\{x=0\,y=\mathrm{\infty }\right\}\right)$

$0$

$\mathrm{limit}\left(xy\,\left\{x=0\,y=\mathrm{\infty }\right\}\right)$

$\mathrm{undefined}$

$\mathrm{limit}\left(\frac{x^3+y^3}{x^2+xy+y^2}\,\left\{x=0\,y=0\right\}\right)$

$0$

$\mathrm{limit}\left(\frac{x^{3}y}{2x^4+y^4}\,\left\{x=0\,y=0\right\}\right)$

$-\frac{6^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\sqrt{2}}{16}..\frac{6^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\sqrt{2}}{16}$

$\mathrm{limit}\left(\frac{x^4-y^2+3x^{2}y-x^2}{x^2+y^2}\,\left\{x=0\,y=0\right\}\right)$

$\mathrm{-1}$

$\mathrm{limit}\left(\frac{\sin \left(x^2\right)-\sin \left(y^2\right)}{x-y}\,\left\{x=0\,y=0\right\}\right)$

$0$

$\mathrm{limit}\left(\frac{1}{y-x}\,\left\{x=y\right\}\right)$

$\mathrm{undefined}$

$\mathrm{limit}\left(\frac{1}{y-x}\,\left\{x=y\right\}\,\mathrm{right}\right)$

$-\mathrm{\infty }$

$\mathrm{limit}\left(\frac{1}{y-x}\,\left\{x=y\right\}\,\mathrm{left}\right)$

$\mathrm{\infty }$

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

The limit/multi command was updated in Maple 2021.