Q. 36 - Calculus

Question

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to 0}{\lim } \frac{x^2-1}{x-1}$.

Step-by-Step Solution

Answer

The value of $\underset{x\to 0}{\lim } \frac{x^2-1}{x-1}$ is, $1.$

Step 1 . Given information

$\underset{x\to 0}{\lim } \frac{x^2-1}{x-1}$.

Step 2 . The limit is calculated as below:

$$\begin{gathered} \underset{x\to 0}{\lim } \frac{x^2-1}{x-1}=\underset{x\to 0}{\lim } \frac{\left(x+1\right)\left(x-1\right)}{(x-1)} \\ =\underset{x\to 0}{\lim } \left(x+1\right) \end{gathered}$$

                   $=\underset{x\to 0}{\lim } \left(x\right)+\underset{x\to 0}{\lim } \left(1\right)$ [Sum rule]

                   $$\begin{gathered} =0+1 \\ =1 \end{gathered}$$