Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 1}\frac{5 x^{2} - 50 x + 45}{5 x^{2} - 2 x - 3}$

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: $LaTeX: \lim_{x \to 1}\frac{5 x^{2} - 50 x + 45}{5 x^{2} - 2 x - 3} = \lim_{x \to 1}\frac{10 x - 50}{10 x - 2} = \frac{10 (1) - 50}{10 (1) - 2} = -5$