Evaluate the limit LaTeX: \displaystyle \lim_{x \to 4}\frac{4 x^{2} - 48 x + 128}{- 7 x^{2} + 24 x + 16}

The limit is an indeterminate form of the type LaTeX: \displaystyle \frac{0}{0} . Using L'Hospitial's rule and then the evaluation theorem gives: LaTeX: \lim_{x \to 4}\frac{4 x^{2} - 48 x + 128}{- 7 x^{2} + 24 x + 16} = \lim_{x \to 4}\frac{8 x - 48}{24 - 14 x} = \frac{8 (4) - 48}{24 - 14 (4)} = \frac{1}{2}