Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 5}\frac{2 x^{2} + 8 x - 90}{7 x^{2} - 27 x - 40}$

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: $LaTeX: \lim_{x \to 5}\frac{2 x^{2} + 8 x - 90}{7 x^{2} - 27 x - 40} = \lim_{x \to 5}\frac{4 x + 8}{14 x - 27} = \frac{4 (5) + 8}{14 (5) - 27} = \frac{28}{43}$