Problem 11 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Limit Problem

We need to find the limit of \( \ln(x+1) \) as \( x \) approaches 0. This means we should analyze the behavior of the natural logarithm function \( \ln \) near \( x = 0 \).

Step 2: Create a Table of Values

To investigate the limit visually, let's create a table of values for \( x \) near 0. Compute \( \ln(x+1) \) for several values of \( x \) such as \( -0.1, -0.01, 0, 0.01, 0.1 \).

Step 3: Calculate Values for the Table

Calculate the following: - \( \ln(-0.1 + 1) = \ln(0.9) \approx -0.105 \)- \( \ln(-0.01 + 1) = \ln(0.99) \approx -0.01005 \)- \( \ln(0 + 1) = \ln(1) = 0 \)- \( \ln(0.01 + 1) = \ln(1.01) \approx 0.00995 \)- \( \ln(0.1 + 1) = \ln(1.1) \approx 0.0953 \)

Step 4: Analyze the Table

Observe the values in the table:- As \( x \to -0 \), \( \ln(x+1) \to -0.01005 \).- At \( x = 0 \), \( \ln(x+1) = 0 \).- As \( x \to +0 \), \( \ln(x+1) \to 0.00995 \).These values suggest that \( \ln(x+1) \rightarrow 0 \) as \( x \to 0 \).

Step 5: Graph the Function

To further confirm, plot the function \( y = \ln(x+1) \) for \( x \) values from \( -0.1 \) to \( 0.1 \). The graph should show a curve passing through \( (0, 0) \), indicating that the function approaches 0 as \( x \) approaches 0.

Step 6: Conclusion

Both the table and graph confirm that as \( x \to 0 \), \( \ln(x+1) \to 0 \). Therefore, the limit is 0.

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln(x) \), is a logarithm to the base \( e \), where \( e \) is a special constant approximately equal to 2.71828. It represents the time needed to reach a certain level of growth. Understanding how \( \ln(x) \) behaves as \( x \) changes is critical, especially when dealing with limits in calculus.
When you see an expression like \( \ln(x+1) \), you're essentially shifting the input to the logarithm function by 1. This shifting can help in analyzing behavior close to certain points, like zero. Near the point where \( x = 0 \), the behavior of \( \ln(x+1) \) is important in determining the limit, as this is the point where small changes in \( x \) have noticeable effects on the output.

Table of Values

Creating a table of values can be a very effective strategy when dealing with limits. It offers a simple way to visually inspect and deduce the behavior of a function as it approaches a specific point. By choosing values incrementally closer to the target point, one can see the trend.
In the exercise here, a table of values for \( x \) near 0 was created and the outputs of \( \ln(x+1) \) were calculated for each. For example:

• \( \ln(0.9) \approx -0.105 \)
• \( \ln(0.99) \approx -0.01005 \)
• \( \ln(1) = 0 \)
• \( \ln(1.01) \approx 0.00995 \)
• \( \ln(1.1) \approx 0.0953 \)

These calculations show that as \( x \) approaches zero from either side, the values of \( \ln(x+1) \) converge towards 0, helping to confirm this limit.

Graphical Analysis

Graphing the function provides another layer of understanding, especially when combined with numerical tables. By plotting \( y = \ln(x+1) \) around the region of interest for \( x \), you can visually confirm how the function behaves.
The graph will show a smooth curve that passes through \( (0,0) \), clearly illustrating that as \( x \) approaches 0, \( y = \ln(x+1) \) also approaches 0. This confirmation through visuals can enhance comprehension by showing the continuous nature of the path and how the values start from negative then approach and finally slightly overshoot zero as \( x \) transitions through point 0.

Continuity and Limits

The concept of limits is fundamental in calculus, as it describes the behavior of functions as they approach a particular point, even if they're not explicitly defined at that point. Continuity, on the other hand, means the function is unbroken and predictable across points.
For the function \( \ln(x+1) \), as \( x \) comes closer to 0, we observe that the output becomes stable around the limit value of 0. Since \( \ln(x+1) \) is defined for all \( x > -1 \) and smoothly connects the values through the point, it is continuous at \( x = 0 \). This property helps reassure that the limit value obtained in this problem is reliable and the function smoothly connects around zero without any jumps or undefined behaviors.