Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -2}\frac{3 x^{2} + 27 x + 42}{- 7 x^{2} - 6 x + 16}$

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: $LaTeX: \lim_{x \to -2}\frac{3 x^{2} + 27 x + 42}{- 7 x^{2} - 6 x + 16} = \lim_{x \to -2}\frac{6 x + 27}{- 14 x - 6} = \frac{6 (-2) + 27}{- 14 (-2) - 6} = \frac{15}{22}$