### Limits And Continuity

(requires JavaScript)

1. Find the limit if it exists or prove that it does not exist.

$\underset{\left(x,y\right)\to \left(5,-2\right)}{\mathrm{lim}}\left({x}^{5}+4{x}^{3}y-5x{y}^{2}\right)$

$2025$
2. Find the limit if it exists or prove that it does not exist.

$\underset{\left(x,y\right)\to \left(0,0\right)}{\mathrm{lim}}\frac{{y}^{4}}{{x}^{4}+3{y}^{4}}$

Does not exist.
3. Find the limit if it exists or prove that it does not exist.

$\underset{\left(x,y\right)\to \left(0,0\right)}{\mathrm{lim}}\frac{6{x}^{3}y}{2{x}^{4}+{y}^{4}}$

The limit along the line $y=0$ is $0$, and the limit along the line $y=x$ is

$\underset{x\to 0}{\mathrm{lim}}\frac{6{x}^{4}}{2{x}^{4}+{x}^{4}}=2$,

so the limit in question does not exist.

4. Determine the set of points where the function

$F\left(x,y\right)=\frac{\mathrm{sin}\left(xy\right)}{{e}^{x}-{y}^{2}}$

is continuous.

$\left\{\left(x,y\right)|y\ne ±{e}^{x/2}\right\}$
5. Determine the set of points where the function

$G\left(x,y\right)=\mathrm{ln}\left({x}^{2}+{y}^{2}-4\right)$

is continuous.

$\left\{\left(x,y\right)|{x}^{2}+{y}^{2}>4\right\}$
6. Use spherical coordinates to find

$\underset{\left(x,y,z\right)\to \left(0,0,0\right)}{\mathrm{lim}}\frac{xyz}{{x}^{2}+{y}^{2}+{z}^{2}}$

$0$