Problem 51 - Calculus for Biology and Medicine

Question

Find the limits in Problems \(51-60 .\) Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow 0} x e^{x} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Write the Expression for the Limit

The initial expression for the limit we need to evaluate is \( \lim_{x \to 0} x e^x \).

Step 2: Check for Indeterminate Forms

Evaluate the expression directly: as \( x \to 0 \), the expression becomes \( 0 \cdot e^0 = 0 \cdot 1 = 0 \). This expression does not directly represent an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Therefore, L'Hôpital's rule is not required.

Step 3: Evaluate the Limit Directly

Since L'Hôpital's rule is not applicable, calculate the limit directly. As \( x \to 0 \), \( e^x \to e^0 = 1 \) and \( x \to 0 \). Therefore, the limit is \( \lim_{x \to 0} x e^x = 0 \cdot 1 = 0 \).

Key Concepts

Limits

In calculus, limits help us understand the behavior of functions as a variable approaches a particular value. This is crucial when the function's formula doesn't provide a clear answer directly.
For example, when we have the limit \( \lim_{x \to 0} x e^x \), it essentially asks, "What value does the expression \( x e^x \) get closer to as \( x \) gets closer to 0?"
Limits are foundational in calculus as they lead to the development of key concepts like derivatives and integrals.

• Approaching a Value: Limits describe what happens to a function as the input gets close to some number. It's not always about where it lands, but where it's headed.
• Handling Uncertainty: Sometimes we get forms like \( \frac{0}{0} \) when evaluating limits, which are hard to interpret directly. This is where limits can help break down that uncertainty.
• Logical Steps: Evaluate functions carefully by plugging in values or using algebraic manipulation.

In our specific example, directly substituting \( x = 0 \) into \( x e^x \) shows us that \(\lim_{x \to 0} x e^x = 0\) because the expression simplifies directly without needing further manipulation.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus for finding limits that result in indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When these forms appear, L'Hôpital's Rule can help us find the limit by differentiating the numerator and the denominator separately.
This method is only applicable when:

• Indeterminate Forms: You encounter expressions that form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
• Derivative Calculation: The derivatives of the numerator and denominator can be calculated easily.

By applying these derivatives, we often simplify the expression into something more manageable to evaluate the limit.
In our initial problem, \( \lim_{x \to 0} x e^x \), we confirmed that L'Hôpital's Rule wasn't needed as it didn’t form an indeterminate \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Instead, evaluating directly gave us a straightforward answer. Understanding when not to use L'Hôpital's Rule is just as crucial as knowing when to apply it.

Indeterminate Forms

Indeterminate forms are expressions involving infinity or zero in limit evaluations. They do not have a straightforward value until further analysis is applied. The most common indeterminate forms are \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \).
When encountering an indeterminate form:

• Look for Patterns: Recognize when a limit leads to these forms.
• Applying Techniques: Use algebraic simplification or calculus tools like L'Hôpital's Rule to resolve the uncertainty.

In some cases, multiplying and dividing by conjugates or other algebraic manipulations can also help.
For our specific exercise, \( \lim_{x \to 0} x e^x \), we tested for indeterminate forms, finding that the expression simply tends to zero as \( x \to 0 \). No further computation or special rules were needed, showing the importance of knowing a variety of techniques.