Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 5}\frac{3 x^{2} - 6 x - 45}{- 7 x^{2} + 44 x - 45}$

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: $LaTeX: \lim_{x \to 5}\frac{3 x^{2} - 6 x - 45}{- 7 x^{2} + 44 x - 45} = \lim_{x \to 5}\frac{6 x - 6}{44 - 14 x} = \frac{6 (5) - 6}{44 - 14 (5)} = - \frac{12}{13}$