Problem 356 - Calculus Volume 1

Question

Evaluate the limit. Evaluate the limit \(\lim _{x \rightarrow \infty} \frac{e^{x}}{x}\).

Step-by-Step Solution

Answer

The limit is \( \infty \).

Step 1: Identify the Type of Limit

We are tasked with evaluating the limit of the function \( f(x) = \frac{e^x}{x} \) as \( x \to \infty \). This is an indeterminate form of \( \frac{\infty}{\infty} \).

Step 2: Apply L'Hôpital's Rule

Since the limit is an indeterminate form \( \frac{\infty}{\infty} \), we apply L'Hôpital's Rule, which states that \( \lim _{x \to \infty} \frac{e^x}{x} = \lim _{x \to \infty} \frac{\frac{d}{dx}(e^x)}{\frac{d}{dx}(x)} \).

Step 3: Differentiate the Numerator and Denominator

The derivative of \( e^x \) is \( e^x \) and the derivative of \( x \) is \( 1 \). Thus, our expression becomes \( \lim _{x \to \infty} \frac{e^x}{1} = \lim _{x \to \infty} e^x \).

Step 4: Evaluate the Reduced Limit

Now, we need to evaluate \( \lim _{x \to \infty} e^x \). As \( x \to \infty \), \( e^x \to \infty \).

Step 5: Conclude the Solution

Therefore, we conclude that \( \lim _{x \to \infty} \frac{e^x}{x} \) equals \( \infty \).

Key Concepts

Limits

Limits are a fundamental concept in calculus, central to understanding the behavior of functions as inputs approach a particular value. In the context of the given exercise, we are interested in the way a function behaves as the variable, in this case, \(x\), trends towards infinity. One common scenario for using limits is finding if a function approaches a finite number or extends to infinity.
If we look at the fraction \( \frac{e^x}{x} \) as \( x \to \infty \), we want to know what value, if any, this expression approaches. When both the numerator and denominator approach infinity, it is classified as an "indeterminate form". This indicates that straightforward substitution is not sufficient, and advanced techniques like L'Hôpital's Rule are required to compute the limit.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool used to resolve indeterminate forms such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), encountered frequently in calculus. The rule provides a way to simplify these expressions by taking derivatives. Specifically, it states that for limits of the form \(\frac{f(x)}{g(x)}\), as \(x\) approaches a value, if directly substituting leads to \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), we can use:

• \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \)

To apply L'Hôpital's Rule for the problem \( \lim_{x \to \infty} \frac{e^x}{x} \), we take the derivatives of the numerator and denominator. The derivative of \(e^x\) is \(e^x\) and the derivative of \(x\) is 1. Thus, the limit changes to \( \lim_{x \to \infty} e^x \), which simplifies the indeterminate form. This calculation shows the function heading towards infinity, confirming the original form's result.

Exponential Functions

Exponential functions, such as \(e^x\), are vital in calculus because of their unique properties, including their rapid growth and their behavior under differentiation and integration. The base of the natural exponential function, \(e\), is an irrational number approximately equal to 2.718. Exponential functions grow faster than polynomial functions, and this characteristic is essential when evaluating limits as \(x\) approaches infinity.
In the specific exercise, the exponential function \(e^x\) in the numerator grows significantly faster than the linear function \(x\) in the denominator. When using L'Hôpital's Rule, we differentiate \(e^x\) and \(x\), finding that while the derivative of the denominator reduces to 1, \(e^x\) remains unchanged, emphasizing the differential growth rates. This is why \( \lim_{x \to \infty} e^x = \infty \), making it clear that exponential growth significantly exceeds that of simple linear functions as \(x\) increases.