Problem 19 - Calculus for Biology and Medicine

Question

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 0^{+}} \frac{1}{\left(1-e^{-x}\right)} $$

Step-by-Step Solution

Answer

The limit is \(-\infty\) as \( x \to 0^{+} \).

Step 1: Understand the Limit Expression

The limit expression given is \( \lim _{x \rightarrow 0^{+}} \frac{1}{\left(1-e^{-x}\right)} \). This means as \( x \) approaches 0 from the positive side, we need to find the behavior of the function \( \frac{1}{1-e^{-x}} \). The key is to understand how the denominator \( 1-e^{-x} \) behaves as \( x \) approaches 0 from the right.

Step 2: Analyze the Denominator

Consider the expression \( e^{-x} \). As \( x \to 0^{+} \), \( e^{-x} \to e^{0} = 1 \). Thus, the denominator \( 1 - e^{-x} \to 1 - 1 = 0 \). Specifically, it approaches 0 from the negative side since \( e^{-x} < 1 \) for all positive values of \( x \).

Step 3: Investigate the Behavior of the Function

Since the denominator approaches 0 from the negative side, \( \frac{1}{1-e^{-x}} \) becomes increasingly negative and large in magnitude as \( x \to 0^{+} \), indicating that the function \( \frac{1}{1-e^{-x}} \to -\infty \).

Step 4: Conclusion Based on Analysis

The function diverges to \(-\infty\) as \( x \to 0^{+} \). Therefore, the limit of \( \frac{1}{1-e^{-x}} \) as \( x \) approaches from the right is \(-\infty\).

Key Concepts

Exponential Functions

Exponential functions are a critical component in many mathematical and real-world applications. These functions have the form \( y = a \cdot e^{bx} \) where \( e \) is the base of the natural logarithm, approximately equal to 2.718. In our exercise, we deal with the exponential expression \( e^{-x} \). This specific form is a decreasing exponential function because the exponent has a negative sign. As \( x \) becomes larger, \( e^{-x} \) becomes smaller, approaching zero. Conversely, as \( x \) approaches zero from the positive side, \( e^{-x} \) approaches 1. This behavior helps in analyzing the limit of the given function as \( x \to 0^+ \). Understanding these properties is essential for studying limits and behaviors involving exponential functions in calculus.

Asymptotic Behavior

Asymptotic behavior examines how functions behave as the input approaches a specific value or infinity. For this exercise, exploring the behavior of \( \frac{1}{1-e^{-x}} \) as \( x \to 0^+ \) involves understanding how the denominator approaches zero.

• As \( x \to 0^+ \), \( e^{-x} \) tends towards 1, making \( 1-e^{-x} \) approach zero from the negative side.
• Given that the denominator approaches zero, the entire fraction increases in magnitude negatively, leading the function towards negative infinity.

This leads to what is called a vertical asymptote at \( x = 0 \). Such asymptotic behavior indicates that as \( x \) gets very close to zero, the function values become indefinitely large in the negative direction.

Graphical Analysis of Limits

Graphical analysis of limits can be particularly insightful for understanding the behavior of functions as they approach a certain point. Graphs provide a visual representation that complements algebraic analysis. In the graph of \( y = \frac{1}{1-e^{-x}} \), as \( x \) approaches 0 from the right, you would observe:

• The values of \( e^{-x} \) climbing towards 1.
• The denominator \( 1-e^{-x} \) shrinking to zero, resulting in larger negative values for the function.
• The function curves steeply downward, confirming the limit of \(-\infty\).

Using graphs to analyze functions helps develop a deeper intuitive understanding of the limit, supporting both the calculations and the conceptual comprehension of behavior such as vertical asymptotes.