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. $$ \lim _{x \rightarrow \infty} \frac{e^{x}}{x} $$

Short answer

You can apply L'Hôpital's rule directly because the limit is \( \frac{\infty}{\infty} \). It evaluates to infinity.

Step-by-step solution

Identify the limit form

Examining the given limit \( \lim_{x \to \infty} \frac{e^x}{x} \), we need to identify the type of indeterminate form. As \( x \to \infty \), both the numerator \( e^x \to \infty \) and the denominator \( x \to \infty \). Therefore, the limit is in the form of \( \frac{\infty}{\infty} \), which is suitable for applying L'Hôpital's rule.

Check for applicability of L'Hôpital's rule

Since we have confirmed that the limit is in an indeterminate form \( \frac{\infty}{\infty} \), we can directly apply L'Hôpital's rule. L'Hôpital's rule states that if the limit \( \lim_{x \to c} \frac{f(x)}{g(x)} \) is in one of the indeterminate forms \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then the limit is equal to \( \lim_{x \to c} \frac{f'(x)}{g'(x)} \), provided the latter limit exists.

Apply L'Hôpital's rule

Calculate the derivatives of the numerator and the denominator. The derivative of the numerator \( e^x \) is \( e^x \), and the derivative of the denominator \( x \) is 1. Therefore, applying L'Hôpital's rule, the limit becomes: \( \lim_{x \to \infty} \frac{e^x}{1} = \lim_{x \to \infty} e^x \).

Evaluate the new limit

The new limit is \( \lim_{x \to \infty} e^x \). As \( x \to \infty \), \( e^x \to \infty \). Therefore, the limit is infinity.

Key concepts

Indeterminate Forms

When evaluating limits, especially as they approach infinity or zero, you might encounter expressions like the one in our exercise: \(\lim_{x \rightarrow \infty} \frac{e^x}{x}\). This expression results in both the numerator and the denominator tending towards infinity, creating the indeterminate form \(\frac{\infty}{\infty}\). Indeterminate forms are expressions that do not lead directly to a specific limit, meaning you cannot immediately determine the limit's value just by looking at them.
This is where calculus tools like L'Hôpital's Rule become essential. Recognizing an indeterminate form is the first step to deciding whether you can proceed with techniques to find the limit. If neither variable approaches zero, infinity, or has more complex relationships, it might not be an indeterminate form.
There are several types of indeterminate forms, including \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0 \cdot \infty\), \(\infty - \infty\), \(0^0\), \(1^\infty\), and \(\infty^0\). Recognizing which kind of expression you have guides you on which calculus techniques you should use to resolve the limit question.

Limit Evaluation

Limit evaluation is the process of determining the value that a function or sequence approaches as the input approaches some value. In this problem, you are tasked with evaluating the limit as \(x\) approaches infinity for the expression \(\frac{e^x}{x}\).
Sometimes, limits appear not easily solvable at first glance. This is especially true for indeterminate forms, where both the numerator and denominator go to infinity or other problematic forms. In such cases, direct substitution will not work, and special techniques, like L'Hôpital's rule, must be used.
The expression \(\frac{e^x}{x}\) becomes \(\frac{\infty}{\infty}\), indicating it is an indeterminate form that aligns well with L'Hôpital's rule application. Evaluating limits helps us understand the behavior of polynomial functions and other complex forms as inputs grow significantly large or small.

Differentiation

Differentiation plays a crucial role when applying L'Hôpital's rule. The rule itself calls for differentiating separately the numerator and the denominator of the expression. For the function \(\frac{e^x}{x}\), you need to find the derivatives of both \(e^x\) and \(x\).
The derivative of \(e^x\) is famously \(e^x\) itself, because the rate of change of \(e^x\) is proportional to the function’s current value. The derivative of the linear function \(x\) is more straightforward, simply 1. By knowing how to differentiate, especially exponential and simple functions, L'Hôpital's rule becomes much more accessible.
Once you compute these derivatives, you substitute back into the limit to evaluate \(\lim_{x \rightarrow \infty} \frac{f'(x)}{g'(x)}\). In this case, the limit becomes \(\lim_{x \rightarrow \infty} e^x\), which leads to a straightforward infinity answer, showing how the growth rate of an exponential function dominates that of a linear function.

Calculus Techniques

Calculus is filled with various techniques for solving problems, and L'Hôpital's rule is a prime example. It provides a structured method for resolving limits involving indeterminate forms like \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\). In the given exercise, you applied this technique directly due to the satisfying conditions of the limit form.
Other calculus techniques exist, such as factoring, conjugates, or even series expansions, but L'Hôpital's rule is often one of the quickest ways to handle suitable indeterminate forms, as it reduces the problem to calculating derivatives. This is especially handy when the original limit setup results in expressions that break traditional rules due to their complex nature.
Using L'Hôpital's rule involves verifying that the expression qualifies as indeterminate, differentiating both numerator and denominator, and finally re-evaluating the new limit. It simplifies the overall limit evaluation by transforming a complex expression into something more recognizable and reducible.