Question

In Problems 33– 44, find the one-sided limit.

$\underset{x\to 1^{-}}{\lim }\frac{x^3-x}{x-1}$

Short answer

$\underset{x\to 1^{-}}{\lim }\frac{x^3-x}{x-1}=2$the solution

Step-by-step solution

step 1.introduction

given limits
$\underset{x\to 1^{-}}{\lim }\frac{x^3-x}{x-1}$

step 2.simplification of equation

given equation

$$\begin{gathered} =\frac{x^3-x}{x-1} \\ =\frac{x(x^2-1)}{x-1} \\ =\frac{x(x+1)(x-1)}{x-1} \\ =x(x+1) \end{gathered}$$

step 3.the solution

the limit value the

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