Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -4}\frac{3 x^{2} + 9 x - 12}{- 9 x^{2} - 38 x - 8}$

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: $LaTeX: \lim_{x \to -4}\frac{3 x^{2} + 9 x - 12}{- 9 x^{2} - 38 x - 8} = \lim_{x \to -4}\frac{6 x + 9}{- 18 x - 38} = \frac{6 (-4) + 9}{- 18 (-4) - 38} = - \frac{15}{34}$