Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 3}\frac{5 x^{2} + 15 x - 90}{- 7 x^{2} + 12 x + 27}$

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: $LaTeX: \lim_{x \to 3}\frac{5 x^{2} + 15 x - 90}{- 7 x^{2} + 12 x + 27} = \lim_{x \to 3}\frac{10 x + 15}{12 - 14 x} = \frac{10 (3) + 15}{12 - 14 (3)} = - \frac{3}{2}$