Question

$\underset{x\to 2}{\lim }\left(\frac{x}{x-2}-\frac{4}{x^2-2x}\right)$

Answer 1

$$\begin{gathered} \underset{x\to 2}{\lim }\left[\frac{x}{x-2}-\frac{4}{x^2-2x}\right] \\ =\underset{x\to 2}{\lim }\left[\frac{x}{x-2}-\frac{4}{x\left(x-2\right)}\right] \\ =\underset{x\to 2}{\lim }\left[\frac{x^2-4}{x\left(x-2\right)}\right] \\ =\underset{x\to 2}{\lim }\left[\frac{\left(x-2\right)\left(x+2\right)}{x\left(x-2\right)}\right] \\ =\underset{x\to 2}{\lim }\left[\frac{\left(x+2\right)}{x}\right] \\ =\frac{2+2}{2} \\ =2 \end{gathered}$$