## Calculus (3rd Edition)

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.