Answer
does not exist
Work Step by Step
See table below.
As x approaches the value 2 from the left,
$f(x)$ becomes a very large in magnitude negative number.
It does not approach any fixed value.
As x approaches the value 2 from the right,
$f(x)$ becomes a very large in magnitude positive number.
It does not approach any fixed value.
If $f(x)$ fails to approach a single fixed number as $x$ approaches $a$ from both sides, then we say that $f(x)$ has no limit as $x\rightarrow a$, or $\displaystyle \lim_{x\rightarrow a}f(x)$ does not exist.
Thus, $\displaystyle \lim_{x\rightarrow 2}\frac{x^{2}-1}{x-2}$ does not exist.