Limit of hard multivariable function

Question

I have this crazy limit that I just can't seem to figure out why the answer is 0. Here is the limit:

$$ \lim_{(x,y) \to (0,0)} (x+y+4)e^{-1/(x^2+y^2)} $$

This evaluates to 0. I would write out my steps but they are pretty pointless considering I always stop when I see that the derivative will give me x or y in denominator giving a undefined answer. Please go step by step because I can do almost any other limit but this one is killing me.

Answer 1

Let $r=\bigl\|(x,y)\bigr\|=\sqrt{x^2+y^2}$. Then both $|x|$ and $|y|$ are smaller thatn or equal to $r$ and therefore$$|x+y+4|\leqslant|x|+|y|+4\leqslant r+4.$$Therefore$$\left|(x+y+4)e^{-\frac1{x^2+y^2}}\right|\leqslant(r+4)e^{-\frac1{r^2}}.$$Since $\lim_{r\to0}(r+4)e^{-\frac1{r^2}}=0$, the limit that you're after is $0$.

Answer 2

$\displaystyle \lim_{(x,y) \to (0,0)} (x+y+4)e^{(-1/(x^2+y^2))}=4\cdot e^{-\infty}=4 \cdot 0=0$