Limits using polar coordinates

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