Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 7}\frac{3 x^{2} - 39 x + 126}{6 x^{2} - 50 x + 56}$

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: $LaTeX: \lim_{x \to 7}\frac{3 x^{2} - 39 x + 126}{6 x^{2} - 50 x + 56} = \lim_{x \to 7}\frac{6 x - 39}{12 x - 50} = \frac{6 (7) - 39}{12 (7) - 50} = \frac{3}{34}$