Problem 86 - Precalculus with Limits

Question

(a) Complete the table for the function $$f(x)=(\ln x) / x$$ (b) Use the table in part (a) to determine what value \(f(x)\) approaches as \(x\) increases without bound. (c) Use a graphing utility to confirm the result of part (b).

Step-by-Step Solution

Answer

The function \( f(x) = (\ln x) / x \) declines as \( x \) increases and seems to approach a certain limit. This limit is confirmed visually by using a graphing utility.

Step 1: Complete the function table

To complete the function table for \( f(x) = (\ln x) / x \), select several values of x to evaluate the function. Remember, the natural logarithm, \( \ln x \), is only defined for \( x > 0 \). So, choose some positive values of \( x \) and calculate the corresponding \( f(x) \) values.

Step 2: Determining limit of \( f(x) \) as \( x \) increases

By observing the values of \( f(x) \) from the table as \( x \) increases, it seems like \( f(x) \) approaches a certain value. The limit this function approaches as \( x \) increases without bound is the value \( f(x) \) seems to be getting closer to as \( x \) increases.

Step 3: Confirming the limit using a graph

Use a graphing utility program to plot the function \( f(x) = (\ln x) / x \). This will give a visual representation of the behavior of \( f(x) \) as \( x \) increases. By observing the graph, you can confirm the limit value determined in Step 2.