Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 1}\frac{3 x^{2} - 30 x + 27}{- 5 x^{2} + 9 x - 4}$

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: $LaTeX: \lim_{x \to 1}\frac{3 x^{2} - 30 x + 27}{- 5 x^{2} + 9 x - 4} = \lim_{x \to 1}\frac{6 x - 30}{9 - 10 x} = \frac{6 (1) - 30}{9 - 10 (1)} = 24$