Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$\lim _{x \rightarrow \infty} \frac{x^{10000}}{e^{x}}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Indeterminate Form

As \( x \) approaches infinity, the numerator \( x^{10000} \) approaches infinity, and the denominator \( e^x \) also approaches infinity. Hence, the expression \( \frac{x^{10000}}{e^{x}} \) is of the indeterminate form \( \frac{\infty}{\infty} \).

Step 2: Apply l'Hôpital's Rule

Since we have an indeterminate form \( \frac{\infty}{\infty} \), we can apply l'Hôpital's Rule, which involves differentiating the numerator and denominator separately. The derivative of the numerator \( x^{10000} \) is \( 10000x^{9999} \), and the derivative of the denominator \( e^x \) is \( e^x \).

Step 3: Reevaluating the Limit

Now our limit becomes: \[ \lim_{x \to \infty} \frac{10000x^{9999}}{e^x} \] As before, this also results in the indeterminate form \( \frac{\infty}{\infty} \), allowing us to apply l'Hôpital's Rule again.

Step 4: Repeated Applications of l'Hôpital's Rule

We continue applying l'Hôpital's Rule. Each application reduces the power of \( x \) in the numerator by one. Thus, the problem can be simplified iteratively by repeatedly differentiating the numerator and denominator until the numerator becomes a constant. This will occur after 10000 applications, resulting in: \( \lim_{x \to \infty} \frac{10000!}{e^x} \).

Step 5: Evaluate the Simplified Limit

Now, evaluate the limit: \[ \lim_{x \to \infty} \frac{10000!}{e^x} = 0 \] because exponential growth in the denominator \( e^x \) eventually overpowers the factorial term in the numerator \( 10000! \).

Key Concepts

Indeterminate Forms

When working with limits, especially as functions approach infinity, you may encounter expressions that are not immediately solvable. These are called indeterminate forms. Indeterminate forms occur because it's unclear what the limit will be based on the expression alone.
The most common types of indeterminate forms encountered are:

• \( \frac{0}{0} \)
• \( \frac{\infty}{\infty} \)
• \( \infty - \infty \)
• \( 0 \times \infty \)
• \( \infty^0 \)
• \( 0^0 \)
• \( 1^\infty \)

For these cases, special tools like l'Hôpital's Rule can help in determining the actual limit. This is why it's important to ensure you first identify an indeterminate form in your limit problem.

Exponential Functions

Exponential functions play a huge role in limit problems, especially as \( x \) approaches infinity. An exponential function is expressed in the form \( e^x \), where \( e \) is Euler's number, approximately 2.71828.
Exponential functions grow extremely fast relative to polynomial functions.
This is a key reason why in many cases, including the given exercise, exponential terms in the denominator will overpower any polynomial terms in the numerator as \( x \to \infty \).
Understanding this will help you predict outcomes when evaluating complex limits. For example, when \( \lim _{x \rightarrow \infty} \frac{x^{10000}}{e^{x}} \) approaches zero, it's because the exponential growth of \( e^x \) is significantly faster than the polynomial growth of \( x^{10000} \).
This makes identifying exponential components in limit problems an important skill.

Limits at Infinity

Limits at infinity help us understand the behavior of functions as the input grows indefinitely.
When we analyze the limit of a function as \( x \to \infty \), we are observing long-term behavior. Different patterns occur with different types of functions, such as exponential and polynomial.

• Polynomial functions, like \( x^{10000} \), continue to increase as \( x \) becomes very large.
• Exponential functions, like \( e^x \), increase even more rapidly and eventually dominate polynomial growth.

When evaluating limits at infinity, it's vital to know which type of function will influence the outcome more significantly. In the example of \( \frac{x^{10000}}{e^{x}} \), because \( e^x \) increases so rapidly, it will force the limit to zero.
Thus, knowing how different functions behave as they approach infinity can clue you into the end result of a limit investigation. Limiting behavior is foundational in calculus and helps in understanding the horizons of mathematical analysis.