Evaluate the limit LaTeX: \displaystyle \lim_{x \to 5}\frac{3 x^{2} - 42 x + 135}{4 x^{2} - 24 x + 20}

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{3 x^{2} - 42 x + 135}{4 x^{2} - 24 x + 20} = \lim_{x \to 5}\frac{6 x - 42}{8 x - 24} = \frac{6 (5) - 42}{8 (5) - 24} = - \frac{3}{4}