another multivarible limits using polar coordinates question

**lllll** · March 25th 2008, 10:16 PM

I'm stuck on

$\lim_{(x,y) \rightarrow (0,0)} \frac{e^{-x^2-y^2}-1}{x^2+y^2}$

turning it into polar coordinates I get:

$\frac{e^{(rcos \theta)^2 - (rsin \theta)^2}-1}{(rcos \theta)^2 + (rsin \theta)^2}$

$=\frac{e^{-r^2}-1}{r^2}$

at which point I get stuck.

**TheEmptySet** · March 25th 2008, 10:53 PM

$\lim_{r \to 0}\frac{e^{-r^2}-1}{r^2}<br />$

using L'hospitals rule we get..

$\lim_{r \to 0}\frac{e^{-r^2}-1}{r^2}=\lim_{r \to 0}\frac{-2re^{r^2}}{2r}$

now taking the limit

$\lim_{r \to 0} -e^{r^2}=-1$

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