Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -7}\frac{3 x^{2} + 30 x + 63}{- 3 x^{2} - 29 x - 56}$

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: $LaTeX: \lim_{x \to -7}\frac{3 x^{2} + 30 x + 63}{- 3 x^{2} - 29 x - 56} = \lim_{x \to -7}\frac{6 x + 30}{- 6 x - 29} = \frac{6 (-7) + 30}{- 6 (-7) - 29} = - \frac{12}{13}$