Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 3}\frac{3 x^{2} + 12 x - 63}{4 x^{2} - 19 x + 21}$

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: $LaTeX: \lim_{x \to 3}\frac{3 x^{2} + 12 x - 63}{4 x^{2} - 19 x + 21} = \lim_{x \to 3}\frac{6 x + 12}{8 x - 19} = \frac{6 (3) + 12}{8 (3) - 19} = 6$