Chapter 4 - Applications of the Derivative - 4.7 L'Hopital's Rule - 4.7 Exercises - Page 307: 13

The solution is $$\lim_{x\to 2}\frac{x^2-2x}{8-6x+x^2}=-1$$

We will evaluate this limit using L'Hopital's rule. "LR" will stand for "Apply L'Hopital's rule" $$\lim_{x\to 2}\frac{x^2-2x}{8-6x+x^2}=\left[\frac{0}{0}\right][\text{LR}]=\lim_{x\to 2}\frac{(x^2-2x)'}{(8-6x+x^2)'}=\lim_{x\to 2}\frac{2x-2}{-6+2x}=\left[\frac{2\cdot 2-2}{-6+2\cdot 2}\right] =-1.$$