Calculate: \lim_{x arrow-2} ((x^2-x-2+|x-2|)/(x^2-4))

Expression: $\lim_{x \rightarrow -2} \left(\frac{ {x}^{2}-x-2+|x-2| }{ {x}^{2}-4 }\right)$

Since the function $\frac{ {x}^{2}-x-2+|x-2| }{ {x}^{2}-4 }$ is undefined for $-2$, evaluate the left-hand and right-hand limits

$\begin{array} { l }\lim_{x \rightarrow -2^-} \left(\frac{ {x}^{2}-x-2+|x-2| }{ {x}^{2}-4 }\right),\\\lim_{x \rightarrow -2^+} \left(\frac{ {x}^{2}-x-2+|x-2| }{ {x}^{2}-4 }\right)\end{array}$

Evaluate the limit

$\begin{array} { l }+\infty,\\\lim_{x \rightarrow -2^+} \left(\frac{ {x}^{2}-x-2+|x-2| }{ {x}^{2}-4 }\right)\end{array}$

Evaluate the limit

$\begin{array} { l }+\infty,\\-\infty\end{array}$

Since the left-hand and the right-hand limits are different, the limit $\lim_{x \rightarrow -2} \left(\frac{ {x}^{2}-x-2+|x-2| }{ {x}^{2}-4 }\right)$ does not exist

$\textnormal{Does not exist}$