Problem 364 - Calculus Volume 1

Question

For the following exercises, 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'Hópital's rule. $$ \lim _{x \rightarrow 0} x^{2 / x} $$

Step-by-Step Solution

Answer

L'Hôpital's rule cannot be applied directly. Transform using logarithms, then use a substitution to apply L'Hôpital's Rule.

Step 1: Identify the limit form

First, we need to recognize the form of the given limit \( \lim_{x \to 0} x^{2/x} \). Typically, L'Hôpital's Rule is applied to limits in the form of \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). The given limit is not in a fraction form as it's an expression of \( x \) raised to the power \( 2/x \).

Step 2: Check if L'Hôpital's Rule Applies Directly

Since \( \lim_{x \to 0} x^{2/x} \) is not in the form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), L'Hôpital's Rule does not directly apply. We need to transform the expression if we want to use L'Hôpital's Rule.

Step 3: Transform the Expression

To transform \( x^{2/x} \) into a form suitable for L'Hôpital's Rule, take the natural logarithm. Let \( y = x^{2/x} \) so that \( \ln(y) = \frac{2}{x} \ln(x) \). Now consider the limit of \( \ln(y) \) as \( x \to 0 \): \( \lim_{x \to 0} \frac{2\ln(x)}{x} \), which is in \( \frac{-\infty}{0} \), indicating more work is required.

Step 4: Examine the Exponent Limit

Convert \( \ln(x) \) to make it amenable to L'Hôpital's: \( \ln(x) \rightarrow -\infty \) as \( x \rightarrow 0 \). Multiply top and bottom by \(-1\) to have \( \lim_{x \to 0} \frac{-2\ln(x)}{x} \), now check if it transforms to \( \frac{-\infty}{\infty} \).

Step 5: Apply L'Hôpital's Rule

Convert problem into \( \frac{\infty}{0} \). Change variables: Let \( x = e^{-t} \), then substitute into \( \lim_{t \to \infty} \frac{-2t}{e^{-t}} \). Now \( e^t \approx \infty \), it's \( \frac{-\infty}{\infty} \), apply L'Hôpital's: derivative of numerator is \(-2\), denominator is \(-t^2 \cdot e^{-t} \), reevaluate to zero.

Key Concepts

Limits in Calculus

Limits are a fundamental concept in calculus, focusing on the behavior of functions as they approach a specific input. They play a crucial role in defining derivatives, integrals, and understanding continuity.
In mathematical analysis, limits help describe the values that a function approaches as the input approaches some point. This is especially useful for examining behavior at points where functions might not be defined.

• Limits can exist as finite numbers, infinity, or not exist at all.
• They are often expressed in formal notation as \( \lim_{x\to a} f(x) = L \).
• This means as \( x \) gets closer to \( a \), \( f(x) \) gets closer to \( L \).

Understanding limits is crucial when dealing with expressions involving continuity and differentiability.
This is because before applying more advanced techniques, like L'Hôpital's Rule, we first need to analyze if a limit exists and what form it takes.

Indeterminate Forms

Indeterminate forms arise when evaluating certain limits that do not initially allow a straightforward calculation. An expression like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) signifies an indeterminate form. At this stage, it is unclear what the limit is, so further manipulation is required.
These forms are significant because they signal ambiguity:

• \( \frac{0}{0} \) means we need to resolve how the numerator and denominator approach zero.
• \( \frac{\infty}{\infty} \) questions how both parts grow, whether one dominates.
• Expressions like \( 0 \cdot \infty \), \( \infty - \infty \), and others also exist.

Identifying indeterminate forms guides whether techniques such as factoring, conjugation, or L'Hôpital's Rule are appropriate.
In our exercise, transforming \( x^{2/x} \) into a fraction was necessary to see if L'Hôpital's Rule could be used for resolving the limit.

Logarithms in Calculus

Logarithms are an essential tool in calculus for simplifying complex functions, especially when working with exponential terms. They transform multiplicative relationships into additive ones, making problems easier to handle.
In the context of L'Hôpital's Rule, logarithms help reformat limits to reveal indeterminate forms.

• Converting exponential expressions. For \( x^{2/x} \), taking the natural log gives us \( \ln(y) = \frac{2}{x} \ln(x) \).
• This allows manipulation using calculus tools like derivatives.
• They also transform the exponential function into a manageable form for limit evaluation.

Specifically, logarithms are crucial when a function is in a power form.We transform it into a logarithmic function, which is then more accessible to techniques like L'Hôpital's Rule for resolving limits effectively.