Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 3}\frac{5 x^{2} + 10 x - 75}{- 8 x^{2} + 23 x + 3}$

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: $LaTeX: \lim_{x \to 3}\frac{5 x^{2} + 10 x - 75}{- 8 x^{2} + 23 x + 3} = \lim_{x \to 3}\frac{10 x + 10}{23 - 16 x} = \frac{10 (3) + 10}{23 - 16 (3)} = - \frac{8}{5}$