Problem 2 - Calculus: Early Transcendentals

Question

Compute the following limits. $$ \lim _{x \rightarrow \infty} \frac{e^{x}}{x^{3}} $$

Step-by-Step Solution

Answer

The limit is infinity.

Step 1: Examine the form

First, examine the limit to identify if it's in an indeterminate form. The expression is \( \frac{e^x}{x^3} \). As \(x\) approaches infinity, \(e^x\) goes to infinity, and \(x^3\) also goes to infinity, resulting in an indeterminate form \( \frac{\infty}{\infty} \).

Step 2: Apply L'Hôpital's Rule

Since the limit is in indeterminate form \( \frac{\infty}{\infty} \), we can use L'Hôpital's Rule. This rule states that if \( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\infty}{\infty} \), then \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \) if this limit exists.Take the derivatives of the numerator and the denominator. \( f(x) = e^x \) gives \( f'(x) = e^x \) and \( g(x) = x^3 \) gives \( g'(x) = 3x^2 \).

Step 3: Re-evaluate the limit

Substitute the derivatives back into the limit:\( \lim_{x \to \infty} \frac{e^x}{3x^2} \). This is still in the indeterminate form \( \frac{\infty}{\infty} \), so we apply L'Hôpital's Rule again.

Step 4: Apply L'Hôpital's Rule again

Differentiate the new numerator and denominator again.Now, \( f(x)= e^x \) gives \( f'(x) = e^x \), and \( g(x) = 3x^2 \) gives \( g'(x) = 6x \).Substitute back into the limit which becomes: \( \lim_{x \to \infty} \frac{e^x}{6x} \).

Step 5: Apply L'Hôpital's Rule one more time

Differentiate the numerator and denominator one more time.With \( f(x) = e^x \), \( f'(x) = e^x \) and \( g(x) = 6x \), \( g'(x) = 6 \).So the limit becomes: \( \lim_{x \to \infty} \frac{e^x}{6} \).

Step 6: Evaluate the final limit

Now, evaluate \( \lim_{x \to \infty} \frac{e^x}{6} \). As \( x \to \infty \), \( e^x \to \infty \), so the whole expression \( \frac{e^x}{6} \) also goes to infinity.Therefore, \( \lim_{x \to \infty} \frac{e^x}{x^3} = \infty \).

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a helpful tool in calculus, especially when dealing with limits of indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). It simplifies the process by allowing us to take the derivatives of the numerator and the denominator in a limit expression.
For mathematicians and students alike, this technique provides a way to reevaluate and find meaningful limits that aren't immediately apparent.
When applying the rule, remember:

• The expressions must initially result in an indeterminate form.
• Taking derivatives must lead to a determinate form where the limit can be found.
• If repeated applications are needed, you may need to apply the rule sequentially multiple times, as is required in infinity cases.

L'Hôpital's Rule is not always applicable, especially if the limit itself doesn't result in an indeterminate form. But when it does apply, it's a powerful method for simplifying complex calculus problems.

Indeterminate Form

In calculus, an indeterminate form refers to an expression involving limits that does not initially reveal any decisive information. Examples of indeterminate forms include \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0 \cdot \infty\), and several others, where simply plugging in the values leads to ambiguity.

These forms necessitate further analysis or alternate mathematical techniques such as L'Hôpital's Rule. It's essential to recognize an indeterminate form as the starting point in determining limits.

• An indeterminate form indicates the need for additional steps to uncover a clear answer.
• It represents situations where the change in both numerator and denominator occurs simultaneously and indefinitely.

Understanding indeterminate forms is vital in calculus since many functions approach such states as variables head toward infinity or finite boundaries, making it crucial to resolve these using appropriate methods.

Exponential Functions

Exponential functions are one of the fundamental building blocks in calculus, characterized by having a constant base raised to a variable exponent, such as \(e^x\). These functions are essential for modeling growth processes and decay in science and economics.

As the input in an exponential function increases, the output typically accelerates swiftly.

• The base of natural logarithms, \(e\), is approximately equal to 2.71828. It is a crucial constant in calculus due to its unique properties, such as having a derivative equal to itself.
• Exponential functions grow much faster than polynomial ones, which means in calculus limits, they often dominate the growth of other parts within the expression.

In solving limits involving exponential functions, always pay attention to their rapid rate of increase which often leads to infinity as the variable tends towards infinity, necessitating careful analytical strategies like L'Hôpital's rule.

Infinity in Calculus

Infinity plays a significant role in calculus, symbolizing values larger than any finite number. Calculating limits that include expressions tending to infinity often leads to understanding the behavior of functions over extensive ranges.

In calculus, you frequently encounter infinity when evaluating limits, either because a variable grows without bound or approaches a value from an infinite standpoint.

• When expressions involve infinity, it is essential to identify whether they are in a form that is determinate or indeterminate.
• Expressions such as \(\frac{\infty}{\infty}\), \(\infty - \infty\), which are indeterminate, require further work to resolve through rules like L'Hôpital's.
• Infinity can also manifest in definite integrals as limits of integration or as asymptotic behavior of functions.

Dealing with infinity requires a solid grasp of calculus concepts, as it's not just about large numbers but understanding asymptotic behavior and results of mathematical analysis.