Limits and Continuity

**Robb** · May 20th 2009, 10:22 PM

Hello MHF,
Needing some help with this question,

What condition must the nonnegative integers $m,n,p$ satisfy to guarantee that $\lim_{(x,y)\to(0,0)} \frac{x^my^n}{(x^2+y^2)^p}$ exists? Prove your answer. [You may assume that $m,n,p$ are not all zero.]

**chisigma** · May 20th 2009, 11:35 PM

With the substitutions...

$x= \rho\cdot \cos \theta$

$y= \rho\cdot \sin \theta$ (1)

... the limit becomes...

$\lim_{(x,y) \rightarrow (0,0)} \frac{x^{m}\cdot y^{n}}{(x^{2} + y^{2})^{p}} = \lim_{\rho \rightarrow 0} \frac{\rho^{m+n}}{\rho^{2p}} (\cos^{m} \theta \cdot \sin^{n} \theta)$ (2)

The limit (2) exist if and only if it is independent from $\theta$ and this is true only if...

$m+n>2\cdot p$ (3)

In this case the limit is $0$ ...

Kind regards

$\chi$ $\sigma$

**HallsofIvy** · May 21st 2009, 04:04 AM

Blast you! You got in before I could say it!

Thread: Limits and Continuity - two variable function

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