Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -5}\frac{3 x^{2} + 6 x - 45}{- 7 x^{2} - 26 x + 45}$

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: $LaTeX: \lim_{x \to -5}\frac{3 x^{2} + 6 x - 45}{- 7 x^{2} - 26 x + 45} = \lim_{x \to -5}\frac{6 x + 6}{- 14 x - 26} = \frac{6 (-5) + 6}{- 14 (-5) - 26} = - \frac{6}{11}$