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Thread: another multivarible limits using polar coordinates question

  1. #1
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    another multivarible limits using polar coordinates question

    I'm stuck on

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

    turning it into polar coordinates I get:

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

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

    at which point I get stuck.
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  2. #2
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    $\displaystyle \lim_{r \to 0}\frac{e^{-r^2}-1}{r^2}
    $

    using L'hospitals rule we get..

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

    now taking the limit

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