Chapter 11.2, Problem 40E

To determine

To find: The value of the $\underset{x\to 2}{\lim }\frac{-\ln \left(x-1\right)}{x-2}$ using table feature of a graphing utility. Also confirm your result graphically.

Answer to Problem 40E

The value of the limits is $\underset{x\to 2}{\lim }\frac{-\ln \left(x-1\right)}{x-2}=-1$

Explanation of Solution

Given information:

Given expression of limit as $\underset{x\to 2}{\lim }\frac{-\ln \left(x-1\right)}{x-2}$

Let, $f\left(x\right)=\frac{-\ln \left(x-1\right)}{x-2}$

The value of the function at $x=2.1$ is,

  $\begin{array}{c}f\left(2.1\right)=\frac{-\ln \left(2.1-1\right)}{2.1-2}\\ =\frac{-\ln \left(1.1\right)}{0.1}\\ =-0.9531\end{array}$

The value of the function at $x=2.01$ is,

  $\begin{array}{c}f\left(2.01\right)=\frac{-\ln \left(2.01-1\right)}{2.01-2}\\ =\frac{-\ln \left(1.01\right)}{0.01}\\ =-0.9950\end{array}$

The value of the function at $x=2.001$ is,

  $\begin{array}{c}f\left(2.001\right)=\frac{-\ln \left(2.001-1\right)}{2.001-2}\\ =\frac{-\ln \left(1.001\right)}{0.001}\\ =-0.99950\end{array}$

The value of the function at $x=2$ is,

  $\begin{array}{c}f\left(2\right)=\frac{-\ln \left(2-1\right)}{2-2}\\ =\frac{0}{0}\\ =\text{undefined}\end{array}$

The value of the function at $x=1.9$ is,

  $\begin{array}{c}f\left(1.9\right)=\frac{-\ln \left(1.9-1\right)}{1.9-2}\\ =\frac{-\ln \left(0.9\right)}{-0.1}\\ =-1.054\end{array}$

The value of the function at $x=1.99$ is,

  $\begin{array}{c}f\left(1.99\right)=\frac{-\ln \left(1.99-1\right)}{1.99-2}\\ =\frac{-\ln \left(0.99\right)}{-0.01}\\ =-1.00503\end{array}$

The value of the function at $x=1.999$ is,

  $\begin{array}{c}f\left(1.999\right)=\frac{-\ln \left(1.999-1\right)}{1.999-2}\\ =\frac{-\ln \left(0.999\right)}{-0.001}\\ =-1.00050\end{array}$

Hence, the complete table is,

  $\begin{array}{cccccccc}x& 2.1& 2.01& 2.001& 2& 1.9& 1.99& 1.999\\ f\left(x\right)& -0.9531& -0.9950& -0.99950& \text{undefined}& -1.054& -1.00503& -1.0005\end{array}$

The value of the limit is, $\underset{x\to 2}{\lim }\frac{-\ln \left(x-1\right)}{x-2}=-1$

Graph of the function can be drawn as,

  

From the graph it is clear that $\underset{x\to 2}{\lim }\frac{-\ln \left(x-1\right)}{x-2}=-1$