Use Polar Coordinates to Find the Limit...

Question

Use polar coordinates to find the limit. [If $(r, \theta)$ are polar coordinates of the point $(x, y)$ with $r \geq 0$, $r \to 0^+$ as $(x,y) \to (0,0)$)]

$$\lim \limits_{(x,y) \to (0,0)} \dfrac{4e^{-x^2-y^2}-4}{x^2+y^2}$$

Answer 1

Hint: use $$\lim_{u\to 0} \frac{\exp u -1}u = 1$$

application: $$ \dfrac{4e^{-x^2-y^2}-4}{x^2+y^2} =4 \dfrac{e^{-r^2}-1}{r^2}\sim_{r\to 0} 4\dfrac{{-r^2}}{r^2} = -4 $$

Answer 2

Transforming the limit to polar gives $\displaystyle \lim_{r \to 0} \frac{4 e^{-r^2}-4}{r^2}$. Use your favorite tools from single-variable calculus to compute from here.