Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 3}\frac{5 x^{2} - 35 x + 60}{8 x^{2} - 21 x - 9}$

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: $LaTeX: \lim_{x \to 3}\frac{5 x^{2} - 35 x + 60}{8 x^{2} - 21 x - 9} = \lim_{x \to 3}\frac{10 x - 35}{16 x - 21} = \frac{10 (3) - 35}{16 (3) - 21} = - \frac{5}{27}$