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

$-\frac{1}{2}..\frac{1}{2}$

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

$0..4$

$f≔\frac{\left|x\right|\+\|y\|}{\sqrt{x^2\+y^2}}\:$

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

$\mathrm{undefined}$

$\genfrac{}{}{0ex}{}{f}{\phantom{y}}|\genfrac{}{}{0ex}{}{\phantom{x}}{x\=0}$

$\frac{\left|y\right|}{\sqrt{y^2}}$

$\mathrm{simplify}\left(\right) assuming \mathrm{real}$

$1$

$\genfrac{}{}{0ex}{}{f}{\phantom{y}}|\genfrac{}{}{0ex}{}{\phantom{x}}{x\=y}$

$\frac{\left|y\right|\sqrt{2}}{\sqrt{y^2}}$

$\mathrm{simplify}\left(\right) assuming \mathrm{real}$

$\sqrt{2}$

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

$0$

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

$\mathrm{\infty }$

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

$0$

$f ≔ \frac{x^2\+y^2\+z^2}{x\+y\+z}\:$

$\mathrm{limit}\left(f\,\left\{x\=0\,y\=0\,z\=0\right\}\right)$

$\mathrm{undefined}$

$\genfrac{}{}{0ex}{}{f}{\phantom{y}}|\genfrac{}{}{0ex}{}{\phantom{x}}{\left[y\=0\,z\=0\right]}$

$x$

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

$0$

$\genfrac{}{}{0ex}{}{f}{\phantom{y}}|\genfrac{}{}{0ex}{}{\phantom{x}}{\left[y\=0\,z\=-x\+x^4\right]}$

$\frac{x^2+{\left(x^4-x\right)}^2}{x^4}$

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

$\mathrm{\infty }$

$f≔\frac{x^2\+y^2\+z^2\+x y z}{\sqrt{x^4\+2 x^2{z}^2\+z^4}\+y^2-{\left|y\right|}^3}\:$

$\mathrm{limit}\left(f\,\left\{x\=0\,y\=0\,z\=0\right\}\right)$

$1$

$\mathrm{simplify}\left(f\right) assuming \mathrm{real}$

$\frac{xyz+x^2+y^2+z^2}{x^2+z^2+y^2-{\left|y\right|}^3}$

$f≔\frac{\sqrt{x^4\+y^4\+z^4}}{x^2\+y^2\+z^2}\:$

$\mathrm{limit}\left(f\,\left\{x\=0\,y\=0\,z\=0\right\}\right)$

$\mathrm{undefined}$

$\genfrac{}{}{0ex}{}{f}{\phantom{y}}|\genfrac{}{}{0ex}{}{\phantom{x}}{\left[y\=0\,z\=0\right]}$

$\frac{\sqrt{x^4}}{x^2}$

$\mathrm{simplify}\left(\right) assuming \mathrm{real}$

$1$

$\genfrac{}{}{0ex}{}{f}{\phantom{y}}|\genfrac{}{}{0ex}{}{\phantom{x}}{\left[y\=x\,z\=x\right]}$

$\frac{\sqrt{3}\sqrt{x^4}}{3x^2}$

$\mathrm{simplify}\left(\right) assuming \mathrm{real}$

$\frac{\sqrt{3}}{3}$

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

$\mathrm{\infty }$

$f ≔\frac{ x y\+x z\+y z}{1-\cos \left( x^2\+y^2\+z^2\right)}\:$

$\mathrm{limit}\left(f\,\left\{x\=0\,y\=0\,z\=0\right\}\right)$

$\mathrm{undefined}$

$\genfrac{}{}{0ex}{}{f}{\phantom{y}}|\genfrac{}{}{0ex}{}{\phantom{x}}{\left[y\=0\,z\=0\right]}$

$0$

$\genfrac{}{}{0ex}{}{f}{\phantom{y}}|\genfrac{}{}{0ex}{}{\phantom{x}}{\left[y\=x\,z\=x\right]}$

$\frac{3x^2}{1-\cos \left(3x^2\right)}$

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

$\mathrm{\infty }$