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Math Help - another multivarible limits using polar coordinates question

  1. #1
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    another multivarible limits using polar coordinates question

    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.
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  2. #2
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    \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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