Problem 27 - Calculus for Biology and Medicine

Question

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty} x^{5} e^{-x} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Indeterminate Form

We need to evaluate the limit \( \lim _{x \rightarrow \infty} x^{5} e^{-x} \). As \( x \rightarrow \infty \), \( x^5 \rightarrow \infty \) and \( e^{-x} \rightarrow 0 \), resulting in the indeterminate form \( \frac{\infty}{\infty} \).

Step 2: Rewrite the Expression

Rewrite the expression as a fraction: \( \lim _{x \rightarrow \infty} \frac{x^5}{e^x} \).

Step 3: Apply L'Hospital's Rule

Since the limit is an indeterminate form \( \frac{\infty}{\infty} \), we apply L'Hospital's Rule by differentiating the numerator and denominator. The first derivative gives us \( \lim _{x \rightarrow \infty} \frac{5x^4}{e^x} \).

Step 4: Repeat L'Hospital's Rule

The limit \( \frac{5x^4}{e^x} \) is still \( \frac{\infty}{\infty} \), so we apply L'Hospital's Rule again: differentiate to get \( \lim _{x \rightarrow \infty} \frac{20x^3}{e^x} \).

Step 5: Continue Applying L'Hospital's Rule

Apply L'Hospital's Rule again, since it's still \( \frac{\infty}{\infty} \): \( \lim _{x \rightarrow \infty} \frac{60x^2}{e^x} \).

Step 6: Yet Again Apply L'Hospital's Rule

Once more, we have \( \frac{\infty}{\infty} \): \( \lim _{x \rightarrow \infty} \frac{120x}{e^x} \).

Step 7: Final Application of L'Hospital's Rule

Finally, use L'Hospital's Rule one last time, obtaining \( \lim _{x \rightarrow \infty} \frac{120}{e^x} \).

Step 8: Evaluate the Final Limit

As \( x \rightarrow \infty \), \( e^x \rightarrow \infty \), so \( \frac{120}{e^x} \rightarrow 0 \). Thus, the limit is 0.

Key Concepts

Limits of Functions

When you explore limits of functions, you're essentially trying to understand the behavior of a function as the input approaches a certain value. In this exercise, we are interested in what happens to the expression \( x^5 e^{-x} \) as \( x \) approaches infinity.

Limits can help identify function trends, such as growth or decay, as an input gets very large or small.

• For example, \( x^5 \) grows larger and larger as \( x \) increases.
• Conversely, \( e^{-x} \) decreases towards zero as \( x \) grows larger.

By analyzing these behaviors, a limit gives us an overall picture of how the function behaves as \( x \) heads off into the distance.

Indeterminate Forms

Indeterminate forms arise when evaluating limits leads to expressions that do not initially provide clear results. These forms are like puzzles within calculus that require special rules or techniques to solve.

In our problem, as \( x \) goes to infinity, we get the form \( \frac{\infty}{\infty} \):

• The numerator, \( x^5 \), results in infinity as \( x \to \infty \).
• The denominator, \( e^x \), also approaches infinity, due to the exponential growth characteristic of \( e^x \).

This represents an indeterminate form. We cannot initially determine the result, hence the need for L'Hospital's Rule, which helps us solve these types of limits by continuing to substitute derivatives until the limit can be evaluated clearly.

Exponential Functions

Exponential functions involve the mathematical constant \( e \) and demonstrate rapid growth or decay. They play a crucial role in calculus, as they often appear in scenarios such as compounding growth or decay processes.

In this exercise, \( e^{-x} \) is an exponential function:

• As \( x \) increases, \( e^{-x} \) decreases exponentially toward zero, showing how quickly exponential decay can occur.
• This tendency to zero is what balances against the polynomial growth of \( x^5 \).

Exponential functions, due to their nature, often lead to indeterminate forms in limits, which makes them interesting subjects in calculus investigations.

Calculus Techniques

Calculus provides essential tools for solving limits and understanding their implications. L'Hospital's Rule is one such technique that directly addresses indeterminate forms, like \( \frac{\infty}{\infty} \), by repeatedly differentiating the numerator and denominator.

Using L’Hospital's Rule, if you encounter an indeterminate form:

• Take the derivative of the top and bottom functions separately.
• Re-evaluate the limit with these new expressions.

In this exercise, we applied L'Hospital's Rule several times until the result was a straightforward expression: \( \frac{120}{e^x} \), which approached zero as \( x \to \infty \).
L'Hospital's Rule demonstrates the power of calculus techniques in simplifying complicated limits, making them manageable and clear to interpret.