Evaluate the limit LaTeX: \displaystyle \lim_{x \to 1}\frac{4 x^{2} - 28 x + 24}{- 3 x^{2} - 2 x + 5}

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{4 x^{2} - 28 x + 24}{- 3 x^{2} - 2 x + 5} = \lim_{x \to 1}\frac{8 x - 28}{- 6 x - 2} = \frac{8 (1) - 28}{- 6 (1) - 2} = \frac{5}{2}