Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 1}\frac{3 x^{2} - 30 x + 27}{x^{2} - 9 x + 8}$

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: $LaTeX: \lim_{x \to 1}\frac{3 x^{2} - 30 x + 27}{x^{2} - 9 x + 8} = \lim_{x \to 1}\frac{6 x - 30}{2 x - 9} = \frac{6 (1) - 30}{2 (1) - 9} = \frac{24}{7}$