Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -3}\frac{2 x^{2} - 12 x - 54}{5 x^{2} + 12 x - 9}$

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: $LaTeX: \lim_{x \to -3}\frac{2 x^{2} - 12 x - 54}{5 x^{2} + 12 x - 9} = \lim_{x \to -3}\frac{4 x - 12}{10 x + 12} = \frac{4 (-3) - 12}{10 (-3) + 12} = \frac{4}{3}$