How to find the multi variable limit Here is the multi variable function:
$$\lim_{(x,y)\to (0,0)} \frac{x^2-2|x|-2|y|}{|x|+|y|}$$
I have no idea to solve the limit
Hint: Use the Squeeze theorem as follows:
$$\color{red}{-2} \le \frac{x^2}{|x|+|y|} - 2 \le \frac{|x|^2}{|x|} -2 = \color{red}{|x| - 2}$$
Since all variables return non-negative results, you can assume $x,y&gt;0$.
Rewriting limit: $\lim_{(x,y)\to (0,0)} \frac{x^2}{x+y} - 2$
Then using polar coordinates $x=rcos\theta$ and $y=rsin\theta$ as $r\to 0$, we get limit is $-2$