multivarible limits using polar coordinates

**lllll** · March 25th 2008, 08:52 PM

I'm stuck on:

$\lim_{(x,y) \rightarrow (0,0)} (x^2+y^2)ln(x^2+y^2)$

so far I have:

$((rcos \theta)^2+(rsin \theta)^2)(ln((rcos \theta)^2+(rsin \theta)^2)$

= $(r^2)(ln(r^2))$

at which point I'm stuck.

**Jen** · March 25th 2008, 09:03 PM

Originally Posted by lllll

I'm stuck on:

$\lim_{(x,y) \rightarrow (0,0)} (x^2+y^2)ln(x^2+y^2)$

so far I have:

$((rcos \theta)^2+(rsin \theta)^2)(ln((rcos \theta)^2+(rsin \theta)^2)$

= $(r^2)(ln(r^2))$

at which point I'm stuck.

Use L'Hospital's rule,

Re-write as

$(r^2)(ln(r^2))=\frac{2ln(r)}{\frac{1}{r^2}}$

By L.H. rule we get,

$\frac{\frac{2}{r}}{\frac{-2}{r^3}}=-r^2$

lim as r goes to zero is zero.

Hope that helps.

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