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 Multivariate Limits

The limit command in Maple 2019 has been enhanced for the case of limits of quotients of multivariate functions:

Many such limits that could not be determined previously are now computable, including all of the following examples.

Returning ranges instead of undefined in the bivariate case

 >
 ${-}\frac{{1}}{{2}}{..}\frac{{1}}{{2}}$ (1)
 >
 ${0}{..}{4}$ (2)

Support for functions containing abs or radicals

 > $f≔\frac{\left|x\right|+|y|}{\sqrt{{x}^{2}+{y}^{2}}}:$
 > $\mathrm{limit}\left(f,\left\{x=0,y=0\right\}\right)$
 ${\mathrm{undefined}}$ (3)

Why?

 > $\genfrac{}{}{0}{}{f}{\phantom{y}}|\genfrac{}{}{0}{}{\phantom{x}}{x=0}$
 $\frac{\left|{y}\right|}{\sqrt{{{y}}^{{2}}}}$ (4)
 >
 ${1}$ (5)
 > $\genfrac{}{}{0}{}{f}{\phantom{y}}|\genfrac{}{}{0}{}{\phantom{x}}{x=y}$
 $\frac{\left|{y}\right|{}\sqrt{{2}}}{\sqrt{{{y}}^{{2}}}}$ (6)
 >
 $\sqrt{{2}}$ (7)
 >
 ${0}$ (8)
 >
 ${\mathrm{\infty }}$ (9)

Support for functions in more than 2 variables

 > $\mathrm{limit}\left(\frac{\mathrm{exp}\left({x}^{3}+{y}^{3}+{z}^{3}\right)-1}{\mathrm{sin}\left({x}^{2}+{y}^{2}+{z}^{2}\right)},\left\{x=0,y=0,z=0\right\}\right)$
 ${0}$ (10)
 >
 > $\mathrm{limit}\left(f,\left\{x=0,y=0,z=0\right\}\right)$
 ${\mathrm{undefined}}$ (11)

Why?

 > $\genfrac{}{}{0}{}{f}{\phantom{y}}|\genfrac{}{}{0}{}{\phantom{x}}{\left[y=0,z=0\right]}$
 ${x}$ (12)
 > $\mathrm{limit}\left(,x=0\right)$
 ${0}$ (13)
 > $\genfrac{}{}{0}{}{f}{\phantom{y}}|\genfrac{}{}{0}{}{\phantom{x}}{\left[y=0,z=-x+{x}^{4}\right]}$
 $\frac{{{x}}^{{2}}{+}{\left({{x}}^{{4}}{-}{x}\right)}^{{2}}}{{{x}}^{{4}}}$ (14)
 > $\mathrm{limit}\left(,x=0\right)$
 ${\mathrm{\infty }}$ (15)
 >
 > $\mathrm{limit}\left(f,\left\{x=0,y=0,z=0\right\}\right)$
 ${1}$ (16)

Why?

 >
 $\frac{{x}{}{y}{}{z}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}}{{{x}}^{{2}}{+}{{z}}^{{2}}{+}{{y}}^{{2}}{-}{\left|{y}\right|}^{{3}}}$ (17)
 > $f≔\frac{\sqrt{{x}^{4}+{y}^{4}+{z}^{4}}}{{x}^{2}+{y}^{2}+{z}^{2}}:$
 > $\mathrm{limit}\left(f,\left\{x=0,y=0,z=0\right\}\right)$
 ${\mathrm{undefined}}$ (18)

Why?

 > $\genfrac{}{}{0}{}{f}{\phantom{y}}|\genfrac{}{}{0}{}{\phantom{x}}{\left[y=0,z=0\right]}$
 $\frac{\sqrt{{{x}}^{{4}}}}{{{x}}^{{2}}}$ (19)
 >
 ${1}$ (20)
 > $\genfrac{}{}{0}{}{f}{\phantom{y}}|\genfrac{}{}{0}{}{\phantom{x}}{\left[y=x,z=x\right]}$
 $\frac{\sqrt{{3}}{}\sqrt{{{x}}^{{4}}}}{{3}{}{{x}}^{{2}}}$ (21)
 >
 $\frac{\sqrt{{3}}}{{3}}$ (22)
 >
 ${\mathrm{\infty }}$ (23)
 >
 > $\mathrm{limit}\left(f,\left\{x=0,y=0,z=0\right\}\right)$
 ${\mathrm{undefined}}$ (24)

Why?

 > $\genfrac{}{}{0}{}{f}{\phantom{y}}|\genfrac{}{}{0}{}{\phantom{x}}{\left[y=0,z=0\right]}$
 ${0}$ (25)
 > $\genfrac{}{}{0}{}{f}{\phantom{y}}|\genfrac{}{}{0}{}{\phantom{x}}{\left[y=x,z=x\right]}$
 $\frac{{3}{}{{x}}^{{2}}}{{1}{-}{\mathrm{cos}}{}\left({3}{}{{x}}^{{2}}\right)}$ (26)
 > $\mathrm{limit}\left(,x=0\right)$
 ${\mathrm{\infty }}$ (27)
 >