Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -7}\frac{5 x^{2} + 50 x + 105}{- 8 x^{2} - 59 x - 21}$

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: $LaTeX: \lim_{x \to -7}\frac{5 x^{2} + 50 x + 105}{- 8 x^{2} - 59 x - 21} = \lim_{x \to -7}\frac{10 x + 50}{- 16 x - 59} = \frac{10 (-7) + 50}{- 16 (-7) - 59} = - \frac{20}{53}$