Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -5}\frac{3 x^{2} + 30 x + 75}{- 3 x^{2} - 16 x - 5}$

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: $LaTeX: \lim_{x \to -5}\frac{3 x^{2} + 30 x + 75}{- 3 x^{2} - 16 x - 5} = \lim_{x \to -5}\frac{6 x + 30}{- 6 x - 16} = \frac{6 (-5) + 30}{- 6 (-5) - 16} = 0$