# Evaluate $\lim\limits_{(x,y)\to(0,0)}\frac{|x|}{|x|+|y|}$

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

• You have not squeezed inasmuch as the left-hand side is not bounded below. – Mark Viola Sep 26 '17 at 21:02
• you haven't "sqeezed" your function, unless you can also find a lower bound, and then show that they are equal to one another. – Doug M Sep 26 '17 at 21:02
• And from the diagonal "axis" (where $x=y \equiv z$): $\lim\limits_{z\to0} {|z| \over |z| + |z|} = 1/2$. – David G. Stork Sep 26 '17 at 22:18