Evaluate the limit LaTeX: \displaystyle \lim_{x \to 4}\frac{5 x^{2} + 10 x - 120}{- 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{5 x^{2} + 10 x - 120}{- 7 x^{2} + 24 x + 16} = \lim_{x \to 4}\frac{10 x + 10}{24 - 14 x} = \frac{10 (4) + 10}{24 - 14 (4)} = - \frac{25}{16}