Chapter 3.4, Problem 62E

To determine

An argument based on the graph of $y=e^x$ to explain why the given statement is false.

Answer to Problem 62E

It has been shown that the given assumption that the graph of $y=\ln x$ looks as though it might be approaching a horizontal asymptote, is false.

Explanation of Solution

Given:

The graph of $y=\ln x$ looks as though it might be approaching a horizontal asymptote.

Concept used:

The graph of the inverse of a function is the reflection of the graph of the given function with respect to the line $y=x$ .

Calculation:

It should be noted that the function $y=e^x$ is inverse of the function $y=\ln x$ .

Then, the graph of $y=e^x$ can be obtained by reflecting the graph of $y=\ln x$ about the line $y=x$ .

If the given assumption that the graph of $y=\ln x$ looks as though it might be approaching a horizontal asymptote, is true, then it must follow that the graph of $y=e^x$ should approach a vertical asymptote.

However, the domain of $y=e^x$ is $ℝ$ and so the graph of $y=e^x$ is defined for all $x\in ℝ$ .

So, it is not possible for the graph of $y=e^x$ to have a vertical asymptote.

Hence, the given assumption that the graph of $y=\ln x$ looks as though it might be approaching a horizontal asymptote, is false.

Conclusion:

It has been shown that the given assumption that the graph of $y=\ln x$ looks as though it might be approaching a horizontal asymptote, is false.