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Math Help - Limits using polar coordinates

  1. #1
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    Limits using polar coordinates

    I am stuck with a question which has to be solved by using polar coordinates to find the limit.

    The question is :

    Find the limit as x and y tend to 0, of the function (1+x^2+y^2)^(1/(x^2+y^2)) using polar coordinates.

    Can someone please help me out with this !
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  2. #2
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    Re: Limits using polar coordinates

    \displaystyle \begin{align*} \lim_{(x, y) \to (0, 0)} \left( 1 + x^2 + y^2 \right)^{\frac{1}{x^2 + y^2}} &= \lim_{r \to 0} \left( 1 + r^2 \right) ^{\frac{1}{r^2}} \\ &= \lim_{ r\to 0}e^{\ln{\left[ \left( 1 + r^2 \right)^{\frac{1}{r^2}} \right]}} \\ &= \lim_{ r \to 0}e^{\frac{\ln{\left( 1 + r^2 \right)}}{r^2}} \\ &= e^{\lim_{r \to 0} \frac{\ln{\left( 1 + r^2 \right)}}{r^2}} \\ &= e^{\lim_{r \to 0}\frac{\frac{2r}{1 + r^2}}{2r}} \textrm{ by L'Hospital's Rule} \\ &= e^{\lim_{r \to 0}\frac{1}{1 + r^2}} \\ &= e^{\frac{1}{1 + 0^2}} \\ &= e \end{align*}
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