Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -3}\frac{2 x^{2} + 24 x + 54}{5 x^{2} + 13 x - 6}$

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: $LaTeX: \lim_{x \to -3}\frac{2 x^{2} + 24 x + 54}{5 x^{2} + 13 x - 6} = \lim_{x \to -3}\frac{4 x + 24}{10 x + 13} = \frac{4 (-3) + 24}{10 (-3) + 13} = - \frac{12}{17}$