Question

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \begin{aligned} &\lim _{x \rightarrow 0} \frac{\ln (x+1)}{x}\\\ &\begin{array}{|l|l|l|l|l|l|l|} \hline x & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\ \hline f(x) & & & & & & \\ \hline \end{array} \end{aligned} $$

Short answer

Based on the calculated function values and observation of function behavior around x=0, the estimated limit of function \( \frac{\ln (x+1)}{x} \) as x approaches 0 is 1.

Step-by-step solution

Evaluate the Function for Given Values of x

Calculate f(x) for each given x-value by substituting x-values into the function \( \frac{\ln (x+1)}{x} \) . Here, f(x) is the y-value corresponding to the x-value.

Completing the Table

After calculating the y-values for each x-values, the table will look like the following: | x | -0.1 | -0.01 | -0.001 | 0.001 | 0.01 | 0.1 | | -: | -: | -: | -: | -: | -: | -: | | f(x) | -1.05170918076 | -1.00501670842 | -1.00050016671 | 1.00050016671 | 1.00501670842 | 1.05170918076 |

Estimating the Limit from the Table

Observe the f(x) values generated above as x approaches 0 from the right and left side. From right side, as x is getting closer to 0, f(x) is getting close to 1 and from the left side, as x is getting closer to 0, f(x) is also getting close to 1. It suggests that the limit of the function as x tends to 0 should be 1.

Confirming Limit from Graph

Graph the function \( \frac{\ln (x+1)}{x} \) using a graphing utility. If the function approaches same y-value from the right and the left sides of x=0, then it confirms your finding in the previous step.