Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -3}\frac{2 x^{2} + 18 x + 36}{8 x^{2} + 27 x + 9}$

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: $LaTeX: \lim_{x \to -3}\frac{2 x^{2} + 18 x + 36}{8 x^{2} + 27 x + 9} = \lim_{x \to -3}\frac{4 x + 18}{16 x + 27} = \frac{4 (-3) + 18}{16 (-3) + 27} = - \frac{2}{7}$