Evaluate the limit LaTeX: \displaystyle \lim_{x \to 3}\frac{3 x^{2} - 30 x + 63}{9 x^{2} - 34 x + 21}

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{3 x^{2} - 30 x + 63}{9 x^{2} - 34 x + 21} = \lim_{x \to 3}\frac{6 x - 30}{18 x - 34} = \frac{6 (3) - 30}{18 (3) - 34} = - \frac{3}{5}