Problem 29 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is 0 as \( x \to 0^+ \).

Step 1: Understanding the Function

The function we need to investigate is \( f(x) = x \ln(x) \). When \( x \to 0^+ \), the function involves \( \ln(x) \), which goes to \(-\infty\) as \( x \to 0^+ \). However, \( x \) multiplies \( \ln(x) \), which goes to 0 as \( x \to 0^+ \). We will explore the behavior of this product.

Step 2: Construct a Table of Values

Create a table of values for \( x \) increasingly closer to 0, such as 0.1, 0.01, 0.001, etc. Calculate the corresponding \( f(x) = x \ln(x) \) for each value of \( x \) to investigate the limit behavior.

Step 3: Analyze the Table

As you fill in the table with values: When \( x = 0.1, f(x) \approx -0.2303 \); when \( x = 0.01, f(x) \approx -0.0461 \); when \( x = 0.001, f(x) \approx -0.0069 \). Notice that as \( x \) gets closer to 0, \( f(x) \) seems to approach 0.

Step 4: Graph the Function

Plot the graph of \( f(x) = x \ln(x) \) for small positive values of \( x \) approaching 0. Observe how the graph behaves. The graph will show that as \( x \) approaches 0, the value of \( f(x) \) approaches 0.

Step 5: Conclusion from Table and Graph

From both the table of values and the graph, we see that as \( x \to 0^+ \), the expression \( x \ln(x) \) approaches 0. Thus, this suggests that the limit exists and is 0.

Step 6: Final Limit Statement

Based on our investigation through the table and graph, we deduce that the limit of \( x \ln(x) \) as \( x \to 0^+ \) is indeed 0.

Key Concepts

Graphing Functions

When talking about calculus and limits, graphing functions is a powerful visual tool. It helps us see the behavior of a function as it approaches a certain point. For the function \(f(x) = x \ln(x)\), creating a graph as \(x\) approaches 0 from the positive side can be very enlightening. To do this, plot the values of \(f(x)\) for small positive values of \(x\). You may start with values like 0.1, 0.01, and 0.001.
As these values get closer to 0, observe the graph. Here are some things to watch for:

• The curve tends to approach the x-axis, suggesting the value of \(f(x)\) is heading toward 0.
• The steep descent in the graph as \(x\) decreases indicates how quickly the logarithmic term influences the product.
• Look for the overall trend of the plot; as \(x\) approaches 0, the graph should clearly suggest a limit of 0.

Graphing allows a quick visual verification for the behavior of functions near specific points, providing a stronger intuition for limit problems.

Table of Values

Using a table of values is like having a data-driven insight into function behavior. For the function \(f(x) = x \ln(x)\), create a table with various values for \(x\) approaching zero from the positive side.
Start with values like 0.1, 0.01, and 0.001, as shown in the step-by-step solution. Calculate each corresponding \(f(x)\):

• At \(x = 0.1\), \(f(x) \approx -0.2303\)
• At \(x = 0.01\), \(f(x) \approx -0.0461\)
• At \(x = 0.001\), \(f(x) \approx -0.0069\)

Notice the pattern: as \(x\) becomes smaller, \(f(x)\) moves closer to zero. Use such table values to hypothesize about the limit behavior. This practical approach of breaking the function down numerically helps in understanding the trend and reinforces the graph findings.

Approaching a Limit

Analyzing how a function behaves as it approaches a particular point is the essence of limits in calculus. For \(f(x) = x \ln(x)\) as \(x \to 0^+\), we are interested in what happens when \(x\) grows infinitesimally small but positive.
Key aspects to consider:

• As \(x\) approaches 0, the term \(\ln(x)\) becomes very large in the negative direction, or \(-\infty\).
• However, multiplying this by \(x\), which nears 0, moderates this effect. Hence, the overall expression is influenced by which term dominates.
• Since \(x\) is a stronger force towards zero in this context, it ensures that the overall product \(x \ln(x)\) also zeroes out.

This demonstrates the delicate balance between parts of a function when calculating limits. The insights from graphing and tabulating values work hand in hand with the analytical approach to solidify the understanding that the limit as \(x \to 0^+\) for this function is indeed 0.