Question

Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically.

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

Short answer

The value of limit of the given function is$\frac{1}{2}$.

Step-by-step solution

Step 1. Given information.

The limit of function here given is,

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

Step 2. Simplify.

$\begin{array}{c}\left(\frac{1}{\ln \left(x\right)}-\frac{1}{x-1}\right):\frac{x-1-\ln \left(x\right)}{\ln \left(x\right)(x-1)}\\ =\underset{x\to 1}{\lim }\left(\frac{x-1-\ln \left(x\right)}{\ln \left(x\right)(x-1)}\right)\end{array}$

Step 3. Apply L’Hopital’s Rule.

Then,

$\begin{array}{c}\underset{x\to 1}{\lim }\left(\frac{-\frac{1}{x}+1}{\frac{x-1}{x}+\ln \left(x\right)}\right)\\ =\underset{x\to 1}{\lim }\left(\frac{\frac{1}{x^2}}{\frac{1}{x^2}+\frac{1}{x}}\right)\end{array}$

Now, plug in the value$x=1$.

$\begin{array}{c}=\frac{\frac{1}{1^2}}{\frac{1}{1^2}+\frac{1}{1}}\\ =\frac{1}{2}\end{array}$

Step 4. Make the table.

Now make a table,

Therefore, on the basis of the values in the table we make the guess that,

$\underset{x\to 1}{\lim }\left(\frac{1}{\ln x}-\frac{1}{x-1}\right)=0.50=\frac{1}{2}$

Step 5. Draw the graph.

Now, as a graphical verification use a graphing device.

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