Question

Use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 1} \frac{\ln x}{x-1} $$

Short answer

The limit of the function as \(x \rightarrow 1 \) is 1.

Step-by-step solution

Graphical Approach and Estimate

First, plot the function \( y = \frac{\ln x}{x-1} \) using a graphing utility. By analyzing the graph, make an initial estimate of the value the function approaches as \( x \) gets close to 1.

Numerical Approach

To reinforce the graphical conclusion, make a table of values, using numbers close to 1 (both smaller and larger than 1). The \(y\)-values corresponding to these \(x\)-values will give a numerical approximation of the limit.

Analytical Approach

To find the exact value of the limit, use L'Hopital's Rule because it is an indeterminate form (\[ \frac{0}{0} \] as \(x\) approaches 1). The derivative of \(\ln x\) is \(\frac{1}{x}\), and the derivative of \(x-1\) is 1, meaning that the limit can be rewritten as \( \lim _{x \rightarrow 1} \frac{1}{x} = 1\).