Problem 366 - Calculus Volume 1

Question

Determine whether you can apply L'Hôpital's rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L'Hôpital's rule. \(\lim _{x \rightarrow \infty} \frac{e^{x}}{x}\)

Step-by-Step Solution

Answer

Apply L'Hôpital's Rule directly because the limit is \( \frac{\infty}{\infty} \) form.

Step 1: Identify the Form of the Limit

First, we need to determine the form of the limit as \( x \to \infty \). Evaluate the behavior of the numerator and denominator separately: The numerator \( e^x \) grows exponentially, while the denominator \( x \) grows linearly. Therefore, the limit is in the form \( \frac{\infty}{\infty} \).

Step 2: Check Applicability of L'Hôpital's Rule

L'Hôpital's Rule can be applied if a limit approaches \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) form. Since our limit is in \( \frac{\infty}{\infty} \) form, we can apply L'Hôpital's Rule to evaluate the limit.

Key Concepts

Limit Evaluation

In calculus, limit evaluation plays a crucial role in understanding the behavior of functions as they approach specific points or infinity. The concept of a limit allows us to analyze how a function behaves and approaches a certain value, which can be critical in various mathematical and applied scenarios. To evaluate a limit effectively, one must:

• Identify the function and point of interest (e.g., as \( x \to \infty \)).
• Analyze the behavior of the numerator and denominator separately, if dealing with a fraction.
• Determine the form of the limit, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), to decide the technique for evaluation.

By accurately evaluating limits, we can derive meaningful insights into the growth, convergence, or divergence of functions, aiding in further mathematical analysis.

Indeterminate Forms

Indeterminate forms arise in limit evaluation when a straightforward computation doesn't yield a definitive result. They signal that more advanced techniques, like L'Hôpital's Rule, are needed. Typical indeterminate forms include:

• \( \frac{0}{0} \)
• \( \frac{\infty}{\infty} \)
• \( 0 \times \infty \)
• \( \infty - \infty \)

When you encounter these forms, it's a sign to dig deeper into the problem. For instance, with the function \( \lim_{x \to \infty} \frac{e^x}{x} \), we have the \( \frac{\infty}{\infty} \) form. This tells us the behavior of the numerator and the denominator both head towards infinity, making direct computation challenging. Here, applying L'Hôpital's Rule allows us to substitute the functions with their derivatives, simplifying the limit evaluation process and leading to a solution.

Exponential Functions

Exponential functions are a key component in calculus, characterized by the expression \( e^x \), where \( e \) is Euler's number, approximately equal to 2.718. These functions grow rapidly compared to polynomial functions, making them particularly influential in limit problems. The growth nature of exponential functions can cause limits to fall into the indeterminate form \( \frac{\infty}{\infty} \), as seen with the limit \( \lim_{x \to \infty} \frac{e^x}{x} \). Unlike linear or polynomial growth, exponential growth outpaces other rates, indicating that directly comparing terms may not yield valuable insight. Using tools like L'Hôpital's Rule becomes essential when handling exponential expressions in calculus. This rule leverages the differentiable nature of exponential functions and their linear counterparts to simplify calculations, making sense of seemingly complex limit evaluations involving exponential behaviors.