Problem 19 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is 0 as \(x\) approaches 0 from the right.

Step 1: Understand the Limit

The problem asks us to find the limit of the function \(1 - e^{-x}\) as \(x\) approaches zero from the positive side. This means we're considering values of \(x\) that are greater than zero but very close to zero.

Step 2: Behavior of \(e^{-x}\) as \(x\) Approaches Zero from the Positive Side

The exponential function \(e^{-x}\) decreases as \(x\) increases. As \(x\) approaches zero from the positive side, \(e^{-x}\) approaches \(e^{-0} = 1\).

Step 3: Substitute into the Function

Given that \(e^{-x}\) approaches 1 as \(x\) approaches zero from the right, substitute this into the function. We have \(1 - e^{-x} = 1 - 1 = 0\).

Step 4: Graphical Interpretation

If you graph the function \(1 - e^{-x}\), you will see that as \(x\) approaches zero from the right, the curve gets closer to zero. This confirms our previous algebraic result.

Step 5: Construct a Table of Values

Create a table of values for \(x\) very close to zero from the positive side. For example, for \(x = 0.1, 0.01, 0.001\), compute \(1 - e^{-x}\): - When \(x = 0.1\), \(1 - e^{-0.1} \approx 0.0952\).- When \(x = 0.01\), \(1 - e^{-0.01} \approx 0.00995\).- When \(x = 0.001\), \(1 - e^{-0.001} \approx 0.0009995\).The results approach zero as \(x\) gets smaller.

Key Concepts

Understanding Exponential Functions

The function we are dealing with in this problem is an exponential function. Exponential functions are special because they involve the constant base \( e \), which is approximately equal to 2.718. This number is unique because it frequently appears in real-world applications, particularly in growth and decay processes in natural phenomena. In this particular exercise, we look at \( e^{-x} \). Here's what you need to know:

• The exponent here is negative, which means as \( x \) becomes larger, \( e^{-x} \) becomes smaller.
• As \( x \) moves towards zero from the positive side, \( e^{-x} \) approaches 1 because \( e^{0} = 1 \).

This behavior can be pivotal in understanding how limits affect exponential functions, especially when dealing with scenarios where \( x \) approaches zero.

Graphical Analysis of Limits

A graphical analysis can provide a clear visual aid to understand the behavior of \( 1 - e^{-x} \) as \( x \) approaches zero. Graphs can make abstract numerical concepts, like limits, more intuitive.When graphing \( 1 - e^{-x} \):

• Plot the curve for values of \( x \) that are close to zero. Notice that the curve rapidly ascends as \( x \) increases due to the subtractive nature of the expression.
• When \( x \) is very near zero, the graph shows the function value is approaching zero, which matches our algebraic findings.

A graph helps reaffirm what our calculations tell us. It shows a decline towards zero, symbolizing how the output of \( 1 - e^{-x} \) diminishes as \( x \) gets closer to zero from the right side.

Using a Table of Values

A table of values is a great way to systematically explore how a function behaves as \( x \) approaches a certain value. In this case, we can see how \( 1 - e^{-x} \) approaches zero as \( x \) nears zero from the positive side.Here's how you can construct and interpret a table of values:

• Choose values for \( x \) that are very small, like 0.1, 0.01, and 0.001.
• For each \( x \), compute \( 1 - e^{-x} \). As seen in the exercise, the results are approximately 0.0952, 0.00995, and 0.0009995, respectively.
• These values indicate that as \( x \) gets smaller (but remains positive), \( 1 - e^{-x} \) gets closer to zero.

Tables give a snapshot of how values trend towards a limit, providing a practical complement to algebraic and graphical analysis.