Problem 366 - Calculus Volume 1

Question

For the following exercises, 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'Hópital's rule. $$ \lim _{x \rightarrow \infty} \frac{e^{x}}{x} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Form

To determine if we can apply L'Hôpital's Rule, we must first identify the limit's form when substituted directly. Substitute \( x \rightarrow \infty \) into \( \frac{e^x}{x} \):\[\frac{e^{x}}{x} \rightarrow \frac{\infty}{\infty}.\]This is an indeterminate form, which means L'Hôpital's Rule can be applied directly.

Step 2: Check Derivatives Condition

L'Hôpital's Rule requires that both the numerator and denominator be differentiable. The differentiation must yield a similar indeterminate form or a resolvable one.\[ \text{Numerator: } e^x, \; \text{Derivative: } e^x \] \[ \text{Denominator: } x, \; \text{Derivative: } 1 \]Both functions are differentiable and their derivatives exist. Since the limit remains in the indeterminate form \( \frac{\infty}{\infty} \), we can apply L'Hôpital's Rule.

Step 3: Apply L'Hôpital's Rule

By applying L'Hôpital's Rule, we differentiate the numerator and the denominator:\[\lim _{x \rightarrow \infty} \frac{e^{x}}{x} = \lim _{x \rightarrow \infty} \frac{\frac{d}{dx} (e^x)}{\frac{d}{dx} (x)} = \lim _{x \rightarrow \infty} \frac{e^x}{1} = \lim _{x \rightarrow \infty} e^x.\]As \( x \rightarrow \infty \), \( e^x \rightarrow \infty \). Thus, the original limit is \( \infty \).

Key Concepts

Indeterminate Forms

Indeterminate forms are mathematical expressions where the limit cannot be immediately determined simply by substituting the point of interest. These forms often occur in calculus when evaluating limits. A classic example is when you plug a value into a function and get forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
These are called indeterminate because they provide no clear information about the actual limit value. Understanding indeterminate forms is crucial when using tools, such as L'Hôpital's Rule, which helps resolve these forms into something tangible. When faced with an indeterminate form like \( \frac{\infty}{\infty} \), as seen in the exercise \( \lim_{x \rightarrow \infty} \frac{e^x}{x} \), we recognize an opportunity to apply L'Hôpital's Rule to find the limit. This rule transforms a complex form into a simpler one by differentiating the numerator and the denominator.

Differentiability

Differentiability is a property of a function that shows how smooth and continuous it is. Specifically, a function is differentiable at a point if it has a derivative there, meaning it behaves predictably and does not create 'jumps' or 'sharp bends' at that point.
For applying L'Hôpital's Rule, it's vital to ensure both the numerator and the denominator are differentiable.
If either function isn't differentiable, we can't safely use the rule. In our step-by-step solution, the numerator \( e^x \) and the denominator \( x \) are both differentiable functions. The derivative of \( e^x \) remains \( e^x \), and the derivative of \( x \) is \( 1 \). Ensuring these derivatives exist and checking they lead to another form, like \( \frac{\infty}{1} \), assures us that we can use L'Hôpital’s Rule properly.

Limits at Infinity

Limits at infinity often help explore the behavior of functions as they grow very large. When evaluating a limit as \( x \) approaches infinity, we are essentially asking what happens to the function values as \( x \) keeps increasing without bound.In such scenarios, if direct substitution returns an indeterminate form, like \( \frac{\infty}{\infty} \), it suggests a more complex interaction between the numerator and denominator.
This complexity often requires additional work, like applying L'Hôpital's Rule to resolve the limit.Using L'Hôpital’s Rule involves taking the derivative of the numerator and the derivative of the denominator. By doing so in our example, the limit \( \lim_{x \rightarrow \infty} \frac{e^x}{1} = \lim_{x \rightarrow \infty} e^x \) shows us that the function value tends toward infinity. Thus, our final result tells us that as \( x \) goes to infinity, \( e^x \) also goes to infinity, providing clear insights into the function’s behavior.