# Thread: Limits and Continuity - two variable function

1. ## Limits and Continuity - two variable function

Hello MHF,
Needing some help with this question,

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

2. With the substitutions...

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

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

... the limit becomes...

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

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

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

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

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