Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 4}\frac{3 x^{2} - 36 x + 96}{- 9 x^{2} + 37 x - 4}$

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: $LaTeX: \lim_{x \to 4}\frac{3 x^{2} - 36 x + 96}{- 9 x^{2} + 37 x - 4} = \lim_{x \to 4}\frac{6 x - 36}{37 - 18 x} = \frac{6 (4) - 36}{37 - 18 (4)} = \frac{12}{35}$