Answer
\[ = \frac{1}{4}\]
Work Step by Step
\[\begin{gathered}
\mathop {\lim }\limits_{x \to \,2} \,\,\,\frac{{\sin \,\left( {x - 2} \right)}}{{{x^2} - 4}} \hfill \\
\hfill \\
factoring\,\,the\,\,denominator\,\, \hfill \\
\hfill \\
= \mathop {\lim }\limits_{x \to \,2} \,\,\,\frac{{\sin \,\,\left( {x - 2} \right)}}{{\,\left( {x + 2} \right)\,\left( {x - 2} \right)}} \hfill \\
\hfill \\
rewriting \hfill \\
= \mathop {\lim }\limits_{x \to \,2} \,\,\,\frac{{\sin \,\,\left( {x - 2} \right)}}{{\,\left( {x + 2} \right)\,\left( {x - 2} \right)}} \hfill \\
\hfill \\
use\,\,the\,\,product\,\,rule\,\,for\,\,\,limits\, \hfill \\
\hfill \\
= \,\left( {\mathop {\lim }\limits_{x \to \,2} \,\,\,\frac{1}{{x + 2}}} \right)\,\left( {\mathop {\lim }\limits_{x \to \,2} \,\,\,\frac{{\sin \,x - 2}}{{x - 2}}} \right) \hfill \\
\hfill \\
evaluate \hfill \\
\hfill \\
= \,\left( {\frac{1}{{2 + 2}}} \right)\,\left( 1 \right) \hfill \\
\hfill \\
= \frac{1}{4} \hfill \\
\hfill \\
\end{gathered} \]
