Question

Use a calculator to graph the function and estimate the value of the limit, then use L'Hôpital's rule to find the limit directly. $$ \lim _{x \rightarrow 0} \frac{e^{x} e^{-x}}{x} $$

Short answer

The limit does not exist.

Step-by-step solution

Simplify the Expression

First, simplify \( e^x e^{-x} \). Notice that \( e^x \) and \( e^{-x} \) are inverse functions, so multiplying them results in 1. Therefore, the limit becomes \( \lim_{x \to 0} \frac{1}{x} \).

Identify the Indeterminate Form

The expression \( \frac{1}{x} \) as \( x \to 0 \) approaches \( \frac{1}{0} \), which is undefined. Therefore, it is an indeterminate form of type \( \frac{1}{0} \).

Analyze Using Graphing Calculator

Use a graphing calculator to plot the function \( \frac{1}{x} \). As \( x \to 0 \), observe that the graph indicates the function is approaching \( \infty \) or \( -\infty \), suggesting no finite limit exists at \( x=0 \).

Apply L'Hôpital's Rule

L'Hôpital's Rule can be applied to limits of the indeterminate forms \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), not \( \frac{1}{0} \). Thus, L'Hôpital's Rule is not applicable directly to \( \lim_{x \to 0} \frac{1}{x} \).

Conclusion from Analysis

Given \( \frac{1}{x} \) approaches infinity as \( x \to 0 \), the limit does not exist in a conventional sense. Rather than applying L'Hôpital's Rule, simply recognize that the function diverges.

Key concepts

Indeterminate Forms

When evaluating limits, some expressions lead to forms that are seemingly undefined or confusing. These are known as indeterminate forms, and they often require special techniques to handle. The classic indeterminate forms are \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\), but there are others like \(0 \cdot \infty\) and even \(1^\infty\). In the context of the exercise, the original limit simplifies to \(\frac{1}{x}\), which at first glance seems like the form \(\frac{1}{0}\). However, strictly speaking, \(\frac{1}{0}\) isn't an indeterminate form in the conventional sense; it’s simply undefined, often leading to infinite behavior rather than necessitating further algebraic manipulation. Recognizing these forms helps determine the best method to evaluate a limit, be it L'Hôpital's Rule or another approach.

Limits in Calculus

Limits are fundamental in calculus for analyzing the behavior of functions as they approach certain points or infinity. They help us understand how functions behave near points of discontinuity or non-differentiability. In the exercise, you have \(\lim_{x \to 0} \frac{1}{x}\), which seeks to investigate the behavior of this function as \(x\) gets very close to zero.
In this scenario, as \(x\) approaches zero from the positive side, \(\frac{1}{x}\) grows increasingly large and positive. Conversely, as \(x\) approaches zero from the negative side, \(\frac{1}{x}\) decreases and becomes large and negative. This type of behavior signifies that the limit does not exist in a traditional sense because the direction from which \(x\) approaches 0 changes the function’s value drastically.
Nonetheless, limits are crucial. They ensure continuity and offer insight into function derivatives and integrals. By understanding how limits work, particularly at problematic points, deeper analyses and interpretations of function behavior become feasible.

Graphing Functions

Graphing functions gives us a visual representation that complements algebraic work when exploring limits. Using a graphing calculator can make it clearer to see certain behaviors like sharp inclines or declines, asymptotes, and other features that are difficult to fully understand through algebra alone.
For the function \(\frac{1}{x}\), graphing shows a hyperbola with a vertical asymptote at the line \(x=0\). As you plot this function, you'll notice two distinct branches: one heading towards positive infinity as \(x\) advances from positive values, and another approaching negative infinity from the negative side. Such a depiction confirms the non-existence of a finite limit.
Utilizing a graphing tool assists in visualizing these changes, allowing for predictions about the function's properties based on its geometric nature. This approach is especially useful when analytical methods prove challenging or complex, providing direct insight into the limit’s behavior near points of discontinuity or undefined nature.