$\begin{array}{l}
\mathop {\lim }\limits_{x \to 4} \dfrac{{{x^2} - 3x - 4}}{{{x^2} - 16}}\\
= \mathop {\lim }\limits_{x \to 4} \dfrac{{\left( {x + 1} \right)\left( {x - 4} \right)}}{{\left( {x - 4} \right)\left( {x + 4} \right)}}\\
= \mathop {\lim }\limits_{x \to 4} \dfrac{{x + 1}}{{x + 4}} = \dfrac{{4 + 1}}{{4 + 4}} = \dfrac{5}{8}
\end{array}$