Question

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \begin{aligned} &\lim _{x \rightarrow 2} \frac{\ln x-\ln 2}{x-2}\\\ &\begin{array}{|l|l|l|l|l|l|l|} \hline x & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & & & & & & \\ \hline \end{array} \end{aligned} $$

Short answer

The numerical limit will be the value that f(x) approaches as x gets increasingly close to 2 from the left and right side, and it will be confirmed visually by observing the graph.

Step-by-step solution

Determine the function values

Use the definition of the function to determine its values at the given x-values. Don't forget to use a standard calculator or a computer algebra system for precise results. The function values will be calculated for x shortly before 2 (1.9, 1.99, 1.999) and shortly after 2 (2.001, 2.01, 2.1).

Fill out the table

Having calculated the function values, fill in the second row of the table.

Estimate the limit

By observing the pattern of the function values, you can estimate the limit. The limit exists if the function approaches a particular value from both the left (values just less than 2) and right hand side (values just greater than 2).

Confirm limit using Graph

Plot the function \(y=\frac{\ln x-\ln 2}{x-2}\) over a suitable range around x=2. Observing the function graphically will either confirm or refute the numerical estimate of the limit.