Problem 52 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is \( +\infty \).

Step 1: Identify the Form

Firstly, identify the form of the limit \( \lim_{x \to 0^{+}} \frac{e^x}{x} \). As \( x \to 0^{+} \), the limit takes the form \( \frac{1}{0^{+}} = \infty \), which is not an indeterminate form of \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), so directly L'Hôpital's rule is not applicable.

Step 2: Evaluate the Limit Directly

Since the form is not indeterminate, evaluate the limit directly. As \( x \to 0^{+} \), \( e^x \rightarrow e^0 = 1 \). Therefore, the expression becomes \( \frac{1}{0^{+}} \), which indicates that the limit approaches \( +\infty \).

Step 3: Conclude the Limit

Conclude that \( \lim_{x \to 0^{+}} \frac{e^x}{x} = +\infty \), since the numerator approaches 1 and the denominator approaches 0 from the positive side, indicating a vertical asymptote.

Key Concepts

L'Hôpital's rule

L'Hôpital's rule is an important mathematical tool used to find limits that result in indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When you encounter these forms, you can apply L'Hôpital's rule to solve them much more easily. Here's how:

• Differentiate the numerator and the denominator of the function separately.
• After differentiating, re-evaluate the limit.
• If the result is still an indeterminate form, you can apply L'Hôpital's rule again.

This rule simplifies problems by reducing the complexity of the expressions involved, making it easier to see the behavior of functions as they approach certain values. However, it's crucial to note that L'Hôpital's rule only applies when the original limit is in an indeterminate form, highlighted as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). If the form isn't indeterminate, like in our example, then L'Hôpital's rule is unnecessary.

Indeterminate Forms

Indeterminate forms are expressions in calculus that do not have a clear or immediate limit. These forms can occur when trying to apply limits, leading to a scenario where each component of the function approaches a limit, but the overall limit is not immediately apparent. The most common indeterminate forms are:

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

When we face a problem like \( \lim_{x \to 0^{+}} \frac{e^x}{x} \), the form wasn't \( \frac{0}{0} \) and came out to be \( \frac{1}{0^{+}} \), a non-indeterminate form, often leading towards infinity. Recognizing indeterminate forms is crucial as it guides the method of solving limit problems, such as determining when to apply L'Hôpital's rule.

Asymptotes

An asymptote is a line that a curve approaches as it heads towards infinity. There are different types of asymptotes, including vertical, horizontal, and oblique. When you evaluate limits, recognizing asymptotes can help determine the behavior of a function as it approaches certain critical points.

• Vertical Asymptotes: These occur when the denominator of a rational function approaches zero, causing the function to go towards infinity, like in our example \( \lim_{x \to 0^{+}} \frac{e^x}{x} = +\infty \).
• Horizontal Asymptotes: Appear when the output of a function approaches a constant value as \( x \) moves towards positive or negative infinity.
• Oblique Asymptotes: Become apparent when a rational function's degree in the numerator is one higher than its degree in the denominator.

Identifying asymptotes is key to understanding the long-term behavior of functions. In our problem, the limit of the function as \( x \to 0^+ \) approaches positive infinity points to a vertical asymptote at \( x = 0 \). This means the function will shoot upwards very sharply as it nears this point.