Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -9}\frac{3 x^{2} + 48 x + 189}{- 4 x^{2} - 31 x + 45}$

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: $LaTeX: \lim_{x \to -9}\frac{3 x^{2} + 48 x + 189}{- 4 x^{2} - 31 x + 45} = \lim_{x \to -9}\frac{6 x + 48}{- 8 x - 31} = \frac{6 (-9) + 48}{- 8 (-9) - 31} = - \frac{6}{41}$