Question

Show that each of the following limits is of the form $\frac{0}{0}$and then use L’Hopital’s rule to calculate the limit: ˆ

Short answer

The solution is 1

Step-by-step solution

Step 1. Given Information 

Given limit is$\underset{x\to 1}{\lim }\frac{\ln \left(x\right)}{x-1}$

Step 2. Calculate the limit

$$\begin{gathered} \underset{x\to 1}{\lim }\frac{\ln \left(x\right)}{x-1} \\ \underset{x\to 1}{\lim }\frac{{\displaystyle \frac{d\left(\ln \left(x\right)\right)}{dx}}}{{\displaystyle \frac{d\left(x-1\right)}{dx}}}=\underset{x\to 1}{\lim }\frac{{\displaystyle \frac{1}{x}}}{1} \\ =\underset{x\to 1}{\lim }\frac{1}{x} \\ =1 \end{gathered}$$

Step 3. The solution 

The solution is 1