Question

Writing to Learn The graph of \(y=\ln x\) looks as though it might be approaching a horizontal asymptote. Write an argument based on the graph of \(y=e^{x}\) to explain why it does not. \([-3,6]\) by \([-3,3]\)

Short answer

The graph of \(y = \ln x\) does not have a horizontal asymptote because, unlike its inverse function \(y = e^{x}\) that approaches 0 (but never reaches it) as \(x\) approaches negative infinity, \(y = \ln x\) approaches negative infinity (and continues to decrease) as \(x\) approaches 0.

Step-by-step solution

Understand the key properties of the function \(y=\ln x\)

The logarithmic function \(y = \ln x\) is the inverse of the exponential function \(y = e^{x}\). This means that the two functions mirror each other across the line \(y = x\). As \(x\) approaches infinity, \(e^{x}\) also approaches infinity and as \(x\) approaches negative infinity, \(e^{x}\) approaches 0 but never reaches it. This means that \(y = e^{x}\) has a horizontal asymptote at \(y = 0\).

Apply the inverse relationship to the graph of \(y=\ln x\)

On the other hand, because \(y = \ln x\) is the inverse of \(y = e^{x}\), if we reflect the graph of \(y = e^{x}\) across the line \(y = x\), we will get the graph of \(y = \ln x\). Hence, the behavior of \(y = \ln x\) as \(x\) approaches 0 is mirrored by the behavior of \(y = e^{x}\) as \(x\) approaches negative infinity. Consequently, \(y = \ln x\) approaches negative infinity as \(x\) approaches 0.

Conclude the absence of a horizontal asymptote in the graph of \(y=\ln x\)

Given that we know, as \(x\) approaches 0, \(y = \ln x\) approaches negative infinity, this implies that there is no horizontal line \(y = c\) where the distance between the curve \(y = \ln x\) and \(y = c\) approaches 0. Therefore, the graph of \(y = \ln x\) does not approach a horizontal asymptote.