1
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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 $p > 1.$

$\endgroup$
3
  • $\begingroup$ 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. $\endgroup$ – user105781 Jan 16 '14 at 2:05
  • $\begingroup$ 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). $\endgroup$ – Viktor Vaughn Jan 16 '14 at 2:26
  • $\begingroup$ I don't think I've learned polar coordinates yet, and i don't understand much of what is going on in this solution. $\endgroup$ – user105781 Jan 16 '14 at 3:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.