# Proof regarding limit with 2 variables

I've encountered a problem which I would like some assistance in doing.

Determine the values of $p$ for which the following limit does or does not exist: $$\lim_{(x,y)\to (0,0)} \frac{\left|x\right|\cdot\left|y\right|^p}{x^2 + y^2}$$

Thanks in advance for any help.

## 1 Answer

Write the expression in polar coordinates, to get

$$r^{p-1} \cos \theta \sin \theta,$$ where $\theta \in [0, \pi/2].$ Now, if $p > 1,$ then the limit is $0$ no matter what $\theta does.$ if $p<1,$ then the limit is infinite along rays where $\theta \in (0, \pi/2),$ and $0$ along the other rays, so the limit does not exist. If $p = 1$ the limits along rays are all finite, but not all the same. So, the limit exits if and only if $p > 1.$

• This isn't really the notation I'm familiar with (polar coordinates) and I honestly do not understand what the "rays" mean. I appreciate the help though. – user105781 Jan 16 '14 at 2:05
• What notation do you use for polar coordinates? I think writing $x = r \cos \theta$ and $y = r \sin \theta$ where $r$ is the radius and $\theta$ is the angle is very standard. A ray is a half-line (cf. ray). – Viktor Vaughn Jan 16 '14 at 2:26
• I don't think I've learned polar coordinates yet, and i don't understand much of what is going on in this solution. – user105781 Jan 16 '14 at 3:54