Answer
Does not exist
Work Step by Step
Given $$\lim _{(x, y) \rightarrow(1,1)} \frac{x^{2}+y^{2}-2}{|x-1|+|y-1|}$$
Let $ u = x − 1$ and $v = y − 1.$ Then
\begin{aligned}
\lim _{(x, y) \rightarrow(1,1)} \frac{x^{2}+y^{2}-2}{|x-1|+|y-1|}=& \lim _{(u, v) \rightarrow(0,0)} \frac{\left((u+1)^{2}+(v+1)^{2}\right)-2}{|u|+|v|} \\
=& \lim _{(u, v) \rightarrow(0,0)} \frac{u^{2}+2 u+v^{2}+2 v}{|u|+|v|} \\
=& \lim _{(u, v) \rightarrow(0,0)} \frac{u^{2}+2 u+v^{2}+2 v}{|u|+|v|}
\end{aligned}
Consider the line $v=mu$ that passes through $(0,0)$:
\begin{align*}
\lim _{(u, v) \rightarrow(0,0)} \frac{u^{2}+2 u+v^{2}+2 v}{|u|+|v|}&=\lim _{u \rightarrow0} \frac{u^{2}+2 u+m^2u^{2}+2mu}{|u|+|m||u|}\\
&=\lim _{u \rightarrow0} \frac{u+2 +m^2u+2m}{1+|m|}\\
&=\frac{ 2 +2m}{1+|m|}
\end{align*}
Since the limit depends on $m$, the limit does not exist.
