Answer
$$\lim _{n \rightarrow 1} \frac{\ln n}{n-1} =1$$
Work Step by Step
Given $$\lim _{n \rightarrow 1} \frac{\ln n}{n-1}$$
using the Limit Rules and replacement, leads to the indeterminate form $$\lim _{n \rightarrow 1} \frac{\ln 1}{1-1}=\frac{0}{0}$$
Applying L'Hôpital's Rule
\begin{align*}
\lim _{n \rightarrow 1} \frac{\ln n}{n-1}&=\lim _{n \rightarrow 1} \frac{\frac{1}{ n}}{1}\\
&= \lim _{n \rightarrow 1} \frac{1}{ n}\\
&= \lim _{n \rightarrow 1} \frac{1}{ 1}\\
&=1
\end{align*}
