Problem 28 - Calculus: Early Transcendentals

Question

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. \( \displaystyle \lim_{x \to 0^+}x^2 \ln x \)

Step-by-Step Solution

Answer

The limit of \(x^2 \ln x\) as \(x\to 0^+\) is \(-\infty\).

Step 1: Understand the Function

The expression we are considering is the limit as \(x\) approaches 0 from the right of \(x^2 \ln x\). "Right of 0" means \(x\) is getting closer to 0 but remains positive.

Step 2: Create a Table of Values

We need to substitute values close to 0 into the expression \(x^2 \ln x\). Let’s choose values that get progressively closer to 0, like \(x = 0.1, 0.01, 0.001,\) and so on.

Step 3: Calculate Values for Each x

Calculate \(x^2 \ln x\) for each of the chosen \(x\) values:For \(x = 0.1\), calculate \((0.1)^2 \ln(0.1)\).For \(x = 0.01\), calculate \((0.01)^2 \ln(0.01)\).For \(x = 0.001\), calculate \((0.001)^2 \ln(0.001)\), and so on.Record these results.

Step 4: Analyze the Table's Results

Observe the results from the table. You will notice that as \(x\) gets closer to 0, the values of \(x^2 \ln x\) become increasingly negative. This pattern suggests that the limit approaches negative infinity.

Step 5: Confirm Graphically

Graph \(x^2 \ln x\) using a graphing calculator or software. Observe the behavior as \(x\) approaches 0 from the right. The graph should show the curve sharply descending towards negative infinity, confirming our table's results.

Key Concepts

Table of Values

When estimating the limit of a function, a useful approach is creating a table of values. This involves selecting values for \( x \) that are increasingly close to the point of interest, which in our example is 0 from the right, hence \( x \to 0^+ \).
To apply this method, pick values like \( x = 0.1, 0.01, 0.001 \), gradually decreasing towards zero. Calculate the expression \( x^2 \ln x \) for each of these \( x \) values.

• For \( x = 0.1 \), compute \( (0.1)^2 \ln(0.1) \).
• For \( x = 0.01 \), compute \( (0.01)^2 \ln(0.01) \).
• Continue this process with \( x = 0.001 \) and other smaller values.

By recording these computations, your table will show how the values of the expression behave as \( x \) inches closer to 0 from the positive side.
Noticing a trend in the results is key. For this function, you observe that as \( x \) approaches 0, the expression \( x^2 \ln x \) yields progressively larger negative values; thus suggesting that the limit approaches negative infinity.

Graphical Confirmation

Graphical confirmation offers a visual approach to understanding the behavior of a function as it approaches a limit. Using a graphing tool or software, you can plot the function \( y = x^2 \ln x \) and observe its behavior as \( x \to 0^+ \).
Focus on the portion of the graph where \( x \) is nearing 0 from the right side. The graph should depict a curve that dips sharply downwards, supporting the conclusion drawn from your table of values.

• If the graph shows the curve approaching negative infinity as \( x \to 0^+ \), it confirms your evaluated limit.
• This graphical observation can provide an additional layer of certainty, especially for functions whose limits seem complex by calculation alone.

Graphical methods are often intuitive and offer a clear representation of how the function behaves, making it an excellent complementary tool to mathematical estimations.

Approaching a Limit from the Right

Approaching a limit from the right (denoted as \( x \to 0^+ \)) involves considering values of \( x \) that come close to the limit point but remain greater than the point itself.
This approach is essential in cases where a function has different behaviors on either side of the point. For example, with \( x^2 \ln x \), it's crucial to only consider positive \( x \) values. This is because \( \ln x \) is undefined for non-positive values.

• In our context, as \( x \to 0^+ \), we examine how the expression \( x^2 \ln x \) behaves specifically from this positive direction.
• By doing so, you get a clear picture of how the function decreases and approaches its limit at negative infinity.

Understanding this concept helps distinguish different limiting behaviors depending on the direction from which \( x \) approaches the limit point, providing a comprehensive view of a function's characteristic near specific values.