Problem 362 - Calculus Volume 1

Question

Determine whether you can apply L'Hôpital's rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L'Hôpital's rule. \(\lim _{x \rightarrow 0^{+}} x^{2} \ln x\)

Step-by-Step Solution

Answer

Rewrite as \( \frac{\ln x}{\frac{1}{x^2}} \) to use L'Hôpital's rule; limit is 0.

Step 1: Understanding L'Hôpital's Rule

L'Hôpital's rule applies for expressions of the form \( \frac{f(x)}{g(x)} \) when the limit results in either \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) indeterminate forms. In this exercise, we have \( x^2 \ln x \), which is a product, not a quotient. Therefore, we cannot directly apply L'Hôpital's rule.

Step 2: Rewriting the Expression

To apply L'Hôpital's rule, we need to rewrite the given expression as a quotient. We express \( x^2 \ln x \) as \( \frac{\ln x}{\frac{1}{x^2}} \). This way, we can examine the limit as \( x \to 0^+ \).

Step 3: Checking Indeterminate Form

We need to check if \( \lim_{x \to 0^+} \frac{\ln x}{\frac{1}{x^2}} \) is an indeterminate form. As \( x \to 0^+ \), \( \ln x \to -\infty \) and \( \frac{1}{x^2} \to \infty \). This creates a \( \frac{-\infty}{\infty} \) indeterminate form, which supports the use of L'Hôpital's rule.

Step 4: Applying L'Hôpital's Rule

Since the rewritten expression \( \frac{\ln x}{\frac{1}{x^2}} \) is in a \( \frac{-\infty}{\infty} \) indeterminate form, we can apply L'Hôpital's rule. Differentiate the numerator and the denominator separately: The derivative of \( \ln x \) is \( \frac{1}{x} \) and the derivative of \( \frac{1}{x^2} \) is \( -\frac{2}{x^3} \). Thus, we need to analyze \( \lim_{x \to 0^+} \frac{\frac{1}{x}}{-\frac{2}{x^3}} = -\frac{1}{2}x^2 \).

Step 5: Solving the Limit After Applying L'Hôpital

Now, evaluate \( \lim_{x \to 0^+} -\frac{1}{2} x^2 \). As \( x \) approaches 0 from the positive side, \( x^2 \to 0 \), thus, \( -\frac{1}{2} x^2 \to 0 \). Therefore, the limit evaluates to 0.

Key Concepts

Indeterminate Forms

In calculus, indeterminate forms arise when evaluating limits that do not initially lead to a clear, defined result. These forms typically occur in the context of limits and express an uncertainty about the value of the limit. The most common indeterminate forms are:

• \( \frac{0}{0} \)
• \( \frac{\infty}{\infty} \)
• \( \infty - \infty \)
• \( 0 \cdot \infty \)
• \( 1^{\infty} \)
• \( 0^{0} \)
• \( \infty^{0} \)

When facing an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), techniques such as L'Hôpital's Rule can be applied to find the limit. In the original exercise, the indeterminate form \( \frac{-\infty}{\infty} \) was encountered after rewriting the expression. This type of indeterminate form suggests that L'Hôpital's Rule may be applied to effectively evaluate the limit.Using differentiation, this rule transforms the form into one that can be more easily evaluated, such as a simple constant or zero, as seen in the exercise solution.

Limits in Calculus

In calculus, limits form the foundation for understanding behaviors of functions as they approach a particular point. They describe the value that a function approaches as the input approaches some value. Limits help in defining derivatives and integrals, making them a core concept in both differential and integral calculus.When working with limits, particularly at points where there might be discontinuities or asymptotic behavior, we rely on defined methodologies and rules. These help ascertain the function's behavior as it approaches particular values.In our specific exercise, we are exploring the limit: \( \lim_{x \to 0^+} x^2 \ln x \). Initially, this expression isn't straightforward in a form that directly reveals its limit. But by rewriting it into a suitable form, we apply limits to understand the behavior more effectively.The importance of limits cannot be overstated, as they underpin key processes and are instrumental in solving problems related to continuity, convergence, and the evaluation of infinite series.

Differentiation Techniques

Differentiation is a primary operation in calculus, involving the computation of derivatives. A derivative represents the rate at which a function's value changes as its input changes. Differentiation provides critical insight into various aspects of functions, such as slope, curvature, and optimization.For L'Hôpital's Rule, differentiation techniques are crucial to transforming and solving limits that exhibit indeterminate forms. This is done by differentiating the numerator and the denominator separately and subsequently reevaluating the limit.In our exercise solution, differentiating \( \ln x \) yields \( \frac{1}{x} \), while differentiating \( \frac{1}{x^2} \) gives \( -\frac{2}{x^3} \). These derivatives help transition from a complex limit into a simpler one, \( -\frac{1}{2}x^2 \), with an easily determined limit as \( x \to 0^+ \).Understanding and applying differentiation techniques not only aid in using L'Hôpital's Rule but also enhance problem-solving abilities across calculus, from analyzing graphs to solving equations and optimizing functions.