Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -1}\frac{2 x^{2} + 6 x + 4}{4 x^{2} - 4}$

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: $LaTeX: \lim_{x \to -1}\frac{2 x^{2} + 6 x + 4}{4 x^{2} - 4} = \lim_{x \to -1}\frac{4 x + 6}{8 x} = \frac{4 (-1) + 6}{8 (-1)} = - \frac{1}{4}$