Limits And Continuity

(requires JavaScript)

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

   $\underset{\left(x,y\right)\to \left(5,-2\right)}{\lim }<!--FUNCTION\; APPLICATION-->\left(x^5+4x^{3}y-5x{y}^2\right)$

   $2025$

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

   ${\displaystyle \underset{\left(x,y\right)\to \left(0,0\right)}{\lim }<!--FUNCTION\; APPLICATION-->\frac{y^4}{x^4+3y^4}}$

   Does not exist.

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

   ${\displaystyle \underset{\left(x,y\right)\to \left(0,0\right)}{\lim }<!--FUNCTION\; APPLICATION-->\frac{6x^{3}y}{2x^4+y^4}}$

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

   ${\displaystyle \underset{x\to 0}{\lim }<!--FUNCTION\; APPLICATION-->\frac{6x^4}{2x^4+x^4}=2}$,

   so the limit in question does not exist.

4. Determine the set of points where the function

   ${\displaystyle F<!--FUNCTION\; APPLICATION-->\left(x,y\right)=\frac{\sin <!--FUNCTION\; APPLICATION-->\left(xy\right)}{e^x-y^2}}$

   is continuous.

   $\left\{\left(x,y\right)|y\ne \pm e^{x/2}\right\}$

5. Determine the set of points where the function

   ${\displaystyle G<!--FUNCTION\; APPLICATION-->\left(x,y\right)=\ln <!--FUNCTION\; APPLICATION-->\left(x^2+y^2-4\right)}$

   is continuous.

   $\left\{\left(x,y\right)|x^2+y^2>4\right\}$