Q. 41 - Calculus

Question

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

$\underset{x\to 1^{+}}{\lim }\frac{x-1}{\sqrt{x-1}}$.

Step-by-Step Solution

Answer

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

Step 1 . Given information

$\underset{x\to 1^{+}}{\lim }\frac{x-1}{\sqrt{x-1}}$.

Step 2 . The limit is calculated as below:

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