Problem 20 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understanding the Limit Expression

The limit we're studying is expressed as \( \lim_{x \to 0} \frac{x}{1 - e^{-x}} \). This limit involves the variable \( x \) approaching 0. Our goal is to evaluate what value this fraction approaches as \( x \) gets closer to 0.

Step 2: Setting Up a Table for Exploration

Create a table with values of \( x \) that approach 0, from both positive and negative directions. Common choices include \(-0.01, -0.001, 0, 0.001, 0.01\). Calculate the corresponding values of the fraction \( \frac{x}{1 - e^{-x}} \) for each value of \( x \).

Step 3: Calculating the Values

Substitute each \( x \) value into the expression \( \frac{x}{1 - e^{-x}} \). For instance, if \( x = 0.001 \), calculate \( 1 - e^{-0.001} \), then divide \( 0.001 \) by this result. Do similarly for all other chosen values of \( x \).

Step 4: Observing the Results

Review the table of results obtained. As \( x \) approaches 0, observe the behavior of the expression \( \frac{x}{1 - e^{-x}} \). Note how the values change as we use smaller values of \( x \) close to 0 from both sides.

Step 5: Interpreting the Limit

As observed, the fraction \( \frac{x}{1 - e^{-x}} \) approaches 1 as \( x \) gets closer to 0. This suggests that the limit is 1. This evaluation is consistent for values approaching from both sides of 0.

Key Concepts

Understanding Limits

In calculus, limits help us understand the behavior of a function as it approaches a particular point. A limit asks what value a function approaches as the input (often denoted as \( x \)), nears a specific number. It's like asking where a car is heading as it gets closer to a particular destination. It's important to remember that in limits, we're not concerned with the value at the exact point, but rather the trend of the function as it nears that point.
When calculating a limit, such as \( \lim_{x \to 0} \frac{x}{1 - e^{-x}} \), we look at \( x \) approaching 0. This means we examine the function's behavior when \( x \) is very near 0, from both the left and the right. This helps us to understand the overall trend and predict its value at that point. Limits are crucial for understanding how functions behave, especially when dealing with complex expressions, like those involving exponential functions.

Exploring Exponential Functions

Exponential functions are a fundamental concept in calculus and mathematics in general. An exponential function is a mathematical expression in the form of \( e^x \), where \( e \) is Euler's number, approximately 2.718. These functions are characterized by their rapid rate of growth or decay.
In our example, \( e^{-x} \) is an exponential function. It describes how rapidly a function can decrease because it's an inverse, thanks to the negative exponent. Exponential functions like these are common in real-world problems, such as modeling populations, radioactive decay, or calculating compound interest, making them highly valuable in various scientific fields.
Understanding exponential functions involves recognizing their behavior and properties, such as:

• The function \( e^x \) increases as \( x \) increases.
• The function \( e^{-x} \) decreases as \( x \) increases.
• Both functions are always positive.

Recognizing these patterns aids you in evaluating expressions that include exponential terms, especially within limits.

The Concept of Approaching Values

Approaching values are essential when analyzing limits. This is about examining how close the function gets to a particular value as the variable approaches a specific point. It's important because sometimes functions may behave unpredictably exactly at the point of interest, but have clear trends leading up to it.
In the example of the limit \( \lim_{x \to 0} \frac{x}{1 - e^{-x}} \), we observe how the expression behaves and what value it approaches as \( x \) gets closer to 0. Here, despite the individual parts of the fraction seeming problematic at \( x = 0 \), the limit reveals a consistent approach toward the value of 1.
This characteristic of approaching helps in:

• Predicting function behavior at points where direct computation is difficult or impossible.
• Identifying trends over specific intervals.
• Preparing for more complex concepts in calculus like derivatives and integrals where limits play a critical role.

By understanding the concept of approaching values, you're better equipped to tackle problems that involve limits and gain deeper insights into the functions you study.