Question

Graphical Analysis In Exercises 61 and 62, graph \(f(x) / g(x)\) and \(f^{\prime}(x) / g^{\prime}(x)\) near \(x=0 .\) What do you notice about these ratios as \(x \rightarrow 0\) ? How does this illustrate L'Hôpital's Rule? \(f(x)=e^{3 x}-1, \quad g(x)=x\)

Short answer

The step by step analysis revealed that as \(x\) approaches 0, both \(f(x)/g(x)\) and \(f'(x)/g'(x)\) approach a similar value. This behavior illustrates L'Hopital's rule, as the ratio of the functions and the ratio of their derivatives have the same limit as \(x\) approaches 0.

Step-by-step solution

Calculating the derivate of each function

The first derivatives of \(f(x)\) and \(g(x)\) are required in order to evaluate \(f'(x) / g'(x)\). Here, we apply derivative rules to compute them : \[f'(x) = 3e^{3x}\] and \[g'(x) = 1\]

Graphing the functions

Using the original function and the derivatives, plot the following ratios near \(x=0\): \[f(x)/g(x)\] and \[f'(x)/g'(x)\]. You can use a graphing calculator or online tool to do this

Observations about the behavior as \(x \rightarrow 0\)

From your graph, note the behavior of the ratios as \(x \rightarrow 0\). Your graph should show that both ratios approach the same value as \(x \rightarrow 0\).

Application of L'Hôpital's Rule

L'Hôpital's rule states that the limit as \(x \rightarrow c\) of \((f(x)-f(c))/(g(x)-g(c))\) is equal to \((f'(x)-f'(c))/(g'(x)-g'(c))\) if the limit exists. In this case, as \(x \rightarrow 0\), the ratios \(f(x)/g(x)\) and \(f'(x)/g'(x)\) approach the same value, demonstrating L'Hôpital's rule.