Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 3}\frac{2 x^{2} - 24 x + 54}{- 4 x^{2} + 21 x - 27}$

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: $LaTeX: \lim_{x \to 3}\frac{2 x^{2} - 24 x + 54}{- 4 x^{2} + 21 x - 27} = \lim_{x \to 3}\frac{4 x - 24}{21 - 8 x} = \frac{4 (3) - 24}{21 - 8 (3)} = 4$