Question

$\underset{x\to 2}{\lim }\left(\frac{1}{x-2}-\frac{4}{x^3-2x^2}\right)$

Answer 1

$$\begin{gathered} \underset{x\to 2}{\lim }\left[\frac{1}{x-2}-\frac{4}{x^3-2x^2}\right] \\ =\underset{x\to 2}{\lim }\left[\frac{1}{x-2}-\frac{4}{x^2\left(x-2\right)}\right] \\ =\underset{x\to 2}{\lim }\left[\frac{x^2-4}{x^2\left(x-2\right)}\right] \\ =\underset{x\to 2}{\lim }\left[\frac{\left(x-2\right)\left(x+2\right)}{x^2\left(x-2\right)}\right] \\ =\underset{x\to 2}{\lim }\left[\frac{x+2}{x^2}\right] \\ =\frac{2+2}{2^2} \\ =1 \end{gathered}$$