Answer
Does not exist
Work Step by Step
Given $$\lim _{(x, y) \rightarrow(0,0)} \frac{|x|}{|x|+|y|}$$
Consider the line $y=mx$ that passes through $(0,0)$:
\begin{align*}
\lim _{(x, y) \rightarrow(0,0)} \frac{|x|}{|x|+|y|}&=\lim _{(x, y) \rightarrow(0,0)} \frac{|x|}{|x|+|mx|}\\
&= \lim _{(x, y) \rightarrow(0,0)} \frac{1}{1+|m|}\\
\end{align*}
Since the limit depends on $m$, then $\lim _{(x, y) \rightarrow(0,0)} \dfrac{|x|}{|x|+|y|}$ does not exist.
