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

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: $LaTeX: \lim_{x \to 1}\frac{2 x^{2} - 4 x + 2}{9 - 9 x^{2}} = \lim_{x \to 1}\frac{4 x - 4}{- 18 x} = \frac{4 (1) - 4}{- 18 (1)} = 0$