Evaluate $\lim\limits_{(x,y)\to(0,0)}\dfrac{|x|}{|x|+|y|}$
The limit does not exist.
From the x-axis (y=0), we have $\lim\limits_{x\to 0} \dfrac{|x|}{|x|+0}=1$
From the y-axis (x=0), we have $\lim\limits_{y\to 0} \dfrac{0}{0+|y|}=0$
Therefore, since $1\neq 0$, the limit does not exist.
But what if we apply the squeeze thereom.
We know that $\dfrac{|x|}{|x|+|y|}\leq \dfrac{|x|}{|x|}=1$, and so the limit is $1$. Why is this incorrect? What are the circumstances that I can use the squeeze thereom, and why isn't this one of them