Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -7}\frac{5 x^{2} + 75 x + 280}{- 9 x^{2} - 69 x - 42}$

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: $LaTeX: \lim_{x \to -7}\frac{5 x^{2} + 75 x + 280}{- 9 x^{2} - 69 x - 42} = \lim_{x \to -7}\frac{10 x + 75}{- 18 x - 69} = \frac{10 (-7) + 75}{- 18 (-7) - 69} = \frac{5}{57}$