Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \lim _{x \rightarrow 0^{+}} \frac{x^{2}}{\sin x-x} $$

Step-by-Step Solution

Answer

The limit approaches \(+\infty\) as \( x \rightarrow 0^+ \).

Step 1: Understand Limits

We need to evaluate \( \lim_{x \to 0^+} \frac{x^2}{\sin x - x} \). This is the right-hand limit as \( x \) approaches 0. Observe that as \( x \to 0 \), both the numerator \( x^2 \) and the expression \( \sin x - x \) in the denominator approaches 0. This is an indeterminate form of \( \frac{0}{0} \), and it suggests we use L'Hôpital's Rule.

Step 2: Apply L'Hôpital's Rule

Since we have a \( \frac{0}{0} \) form, L'Hôpital's Rule applies: \( \lim_{x \to 0^+} \frac{f(x)}{g(x)} = \lim_{x \to 0^+} \frac{f'(x)}{g'(x)} \) if the limit exists. Here, \( f(x) = x^2 \) and \( g(x) = \sin x - x \).

Step 3: Differentiate the Numerator and Denominator

Differentiate \( f(x) = x^2 \), giving \( f'(x) = 2x \). Differentiate \( g(x) = \sin x - x \), giving \( g'(x) = \cos x - 1 \).

Step 4: Evaluate the Limit of Derivatives

Now consider \( \lim_{x \to 0^+} \frac{f'(x)}{g'(x)} = \lim_{x \to 0^+} \frac{2x}{\cos x - 1} \). Again, as \( x \to 0 \), both the numerator and denominator approach 0 \((\frac{0}{0})\). Apply L'Hôpital's Rule again.

Step 5: Apply L'Hôpital's Rule Again

Differentiate \( 2x \) to get \( 2 \) and \( \cos x - 1 \) to get \( -\sin x \). Evaluate the new limit: \( \lim_{x \to 0^+} \frac{2}{-\sin x} = \lim_{x \to 0^+} \frac{-2}{\sin x} \).

Step 6: Find the Final Limit

As \( x \to 0^+ \), \( \sin x \to 0 \), making the limit approach \( -\infty \). This indicates that the original limit approaches \( +\infty \) since as we approach from the right \( \lim_{x \to 0^+} \frac{-2}{\sin x} \rightarrow -\infty \). Therefore, the given limit evaluates to \( +\infty \).

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is an important technique in calculus used to find limits of indeterminate forms. When directly substituting a point into a limit yields an undefined form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), L'Hôpital's Rule can help solve it. The rule states that if \( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \) given that the limit on the right exists. This means we differentiate the numerator and the denominator separately, then find the limit of the new fraction. It's important to note that L'Hôpital's Rule can be applied repeatedly if the resulting form is still indeterminate.

Indeterminate Forms

Limits often lead to what are called indeterminate forms. These forms such as \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \cdot \infty \), \( \infty - \infty \), and others, do not readily provide a specific result or direction. This is why special techniques like L'Hôpital's Rule are employed to resolve them. When you encounter an indeterminate form, it signals the need for further analysis to compute the limit properly. The purpose here is to manipulate or transform the existing problem into one with a determinate form which can be evaluated directly.

Calculus Problem-solving

Solving calculus problems often involves a strategic approach for breaking down complex expressions. Here is a general approach:

• Understand the Problem: Start by clearly defining what the problem is asking. Check for any possible indeterminate forms.
• Choose the Right Method: Selection of the correct technique such as L'Hôpital's Rule, algebraic manipulation, or factoring is crucial.
• Step-by-step Approach: Solve the problem step by step without skipping any transformations or calculations.
• Verify Results: Always plug back your solution or use an alternative method to verify the results.

Incremental solving is essential to ensure no small detail is overlooked, given the sequential steps often required in calculus problem solving.

Differentiation

Differentiation is a cornerstone concept in calculus that refers to finding the derivative of a function. Derivatives measure how a function changes as its input changes, and they are represented by \( f'(x) \) or \( \frac{dy}{dx} \). In practical terms, the derivative gives us the rate of change of one variable with respect to another. Differentiation is used in L'Hôpital's Rule to transform an indeterminate form into one that can be evaluated. This is done by finding the derivatives of the given functions in the numerator and the denominator. Each differentiation step should be done carefully to ensure the correct transformation from the original limit expression.