Problem 356 - Calculus Volume 1

Question

For the following exercises, evaluate the limit. Evaluate the limit \(_{x \rightarrow \infty} \frac{e^{x}}{x}\)

Step-by-Step Solution

Answer

The limit is \(\infty\).

Step 1: Identify the Form

First, we recognize the limit to evaluate: \( \lim_{x \to \infty} \frac{e^x}{x} \). This limit involves an exponential function \(e^x\) and a linear function \(x\).

Step 2: Analyze the Growth of Functions

Compare how functions grow as \(x\) approaches infinity. The exponential function \(e^x\) grows much faster than the linear function \(x\).

Step 3: Apply Limits to Compare Growth Rates

Apply limits to formally see this growth difference. Since \(e^x\) in the numerator grows much larger than \(x\) in the denominator, the fraction \(\frac{e^x}{x}\) will continue to grow towards infinity as \(x\) increases.

Step 4: Use L'Hôpital's Rule (Optional Verification)

If needed, use L'Hôpital's Rule, which is applicable for indeterminate forms \(\frac{\infty}{\infty}\). Compute the derivatives: the derivative of \(e^x\) is \(e^x\), and the derivative of \(x\) is 1. Apply the rule: \(\lim_{x \to \infty} \frac{e^x}{1} = \lim_{x \to \infty} e^x = \infty\).

Step 5: Conclude the Limit

Thus, since \(\frac{e^x}{x}\) grows indefinitely large, \(\lim_{x \to \infty} \frac{e^x}{x} = \infty\).

Key Concepts

Exponential Growth

Exponential growth refers to a situation where a quantity increases rapidly over time. In terms of mathematics, an exponential function like \( e^x \) grows faster than polynomial or linear functions as \( x \) approaches infinity. This rapid increase happens because exponential functions involve raising a constant base, such as Euler's number \( e \), to the power of an ever-increasing exponent.

• The base \( e \) is an important constant, approximately equal to 2.71828.
• As \( x \) becomes very large, \( e^x \) increases dramatically.
• This rapid growth is key to understanding why \( \frac{e^x}{x} \) becomes infinitely large as \( x \) approaches infinity.

When comparing the growth rates of different types of functions, the exponential function outpaces polynomial functions because of this inherent property. This means that as \( x \) becomes large, \( e^x \) dwarfs linear functions such as \( x \), making the multiplying factor for the limit go to infinity as well.

L'Hôpital's Rule

L'Hôpital's Rule is a mathematical method used to resolve indeterminate forms, typically those in the form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

• This rule is applicable if, after direct substitution into a limit, the expression results in one of these indeterminate forms.
• To apply it, take the derivative of the numerator and the derivative of the denominator separately.
• Re-evaluate the limit using the new fraction formed by these derivatives.

In our example, \( \lim_{x \to \infty} \frac{e^x}{x} \), we find an indeterminate form \( \frac{\infty}{\infty} \). By applying L'Hôpital's Rule, we compute the derivative of \( e^x \) (which remains \( e^x \)) and the derivative of \( x \) (which is 1). Substituting back into the limit gives \( \lim_{x \to \infty} \frac{e^x}{1} = \lim_{x \to \infty} e^x = \infty \). Thus, L'Hôpital's rule helps confirm that the limit indeed approaches infinity.

Indeterminate Forms

Indeterminate forms occur in calculus when the limit results in a form that doesn't directly suggest a specific result. For example, when evaluating \( \lim_{x \to \infty} \frac{e^x}{x} \), direct substitution gives the indeterminate form \( \frac{\infty}{\infty} \).

• This means both the numerator and denominator grow indefinitely large and it isn't immediately clear what the behavior of the fraction will be.
• There are several indeterminate forms, including \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \times \infty \), and others.
• Identifying an expression as an indeterminate form is the first step in applying techniques like ratios of derivatives, known as L'Hôpital's Rule, to resolve its behavior.

Understanding these forms is crucial for correctly applying methods like L'Hôpital's Rule to find limits. In this case, since \( e^x \) grows much faster than \( x \), despite both tending towards infinity, the result is that the fraction itself becomes infinitely large.