Proving limits of multivariate function with epsilon-delta definition

Question

I want to solve this problem using epsilon-delta definition :

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

However, I have no idea where to start first - how to modify and apply some kind of inequality.

Answer 1

For $r \ge 1$, you have $1 \le \frac{e^r}{r^2}$ hence $0 < e^{-r}\le \frac{1}{r^2}$.

Replace $r$ with $1/(x^2+y^2)$. You get

$$e^{-1/(x^2+y^2)} \le (x^2+y^2)^2$$ and therefore

$$0 < \frac{e^{-1/(x^2+y^2)}}{x^2+y^2}\le (x^2+y^2)$$ for

$\sqrt{x^2+y^2} \le 1$. Now, fix $\epsilon > 0$ and take $\delta = \min(1, \sqrt{\epsilon})$. If $\sqrt{x^2+y^2} \le \delta$, you get

$$0 < \frac{e^{-1/(x^2+y^2)}}{x^2+y^2}\le \epsilon$$