Evaluate the limit LaTeX: \displaystyle \lim_{x \to 1}\frac{3 x^{2} - 27 x + 24}{5 x^{2} - 8 x + 3}

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} - 27 x + 24}{5 x^{2} - 8 x + 3} = \lim_{x \to 1}\frac{6 x - 27}{10 x - 8} = \frac{6 (1) - 27}{10 (1) - 8} = - \frac{21}{2}