Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{\tan ^{-1} x-x}{8 x^{3}} $$

Step-by-Step Solution

Answer

The limit is \(-\frac{1}{24}\).

Step 1: Identify the Indeterminate Form

First, substitute 0 into the limit expression \( \frac{\tan^{-1} x - x}{8x^3} \). We have \( \tan^{-1}(0) = 0 \) and thus the expression becomes \( \frac{0 - 0}{8 \cdot 0^3} = \frac{0}{0} \), an indeterminate form. Since we have confirmed the indeterminate form, l'Hôpital's Rule can be applied.

Step 2: Apply l'Hôpital's Rule for the First Derivative

According to l'Hôpital's Rule, we derive the numerator and the denominator separately. The derivative of the numerator \( \tan^{-1} x - x \) is \( \frac{1}{1+x^2} - 1 \). The derivative of the denominator \( 8x^3 \) is \( 24x^2 \). The new limit becomes:\[ \lim_{x \to 0} \frac{\frac{1}{1+x^2} - 1}{24x^2} \]

Step 3: Confirming Indeterminate Form Again

Substitute 0 into the new limit expression. The numerator becomes \( \frac{1}{1+0^2} - 1 = 0 \) and the denominator is \( 24 \cdot 0^2 = 0 \). Therefore, the form remains \( \frac{0}{0} \), indicating a need to apply l'Hôpital's Rule again.

Step 4: Apply l'Hôpital's Rule for the Second Derivative

Further differentiate the numerator \( \frac{1}{1+x^2} - 1 \). The derivative is \( \left(-\frac{2x}{(1+x^2)^2}\right) \). The derivative of the denominator \( 24x^2 \) is \( 48x \). The new limit becomes:\[ \lim_{x \to 0} \frac{-\frac{2x}{(1+x^2)^2}}{48x} \]

Step 5: Simplify the Expression

The expression \( \frac{-\frac{2x}{(1+x^2)^2}}{48x} \) can be simplified. Cancelling \( x \) from the numerator and denominator, we get:\[ \lim_{x \to 0} \frac{-2}{48(1+x^2)^2} \]

Step 6: Evaluate the Limit

Now substitute \( x = 0 \) into the simplified expression:\[ \lim_{x \to 0} \frac{-2}{48(1+0^2)^2} = \frac{-2}{48} \]Simplifying this fraction gives \( \frac{-1}{24} \).

Key Concepts

Indeterminate Forms

In calculus, an indeterminate form arises when a mathematical expression results in a form like \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), or similar ambiguous forms when substituting a limit value. These forms are termed indeterminate because they don't lead directly to a specific numerical outcome. Hence, further analysis is needed to find the actual limit. In the original exercise, substituting \(0\) in the expression \(\frac{\tan^{-1} x - x}{8x^3}\) initially results in a \(\frac{0}{0}\) form. This is a classic example of an indeterminate form. Recognizing such forms is crucial as they often indicate where l'Hôpital's Rule can be employed to evaluate the limit properly.

Limit Evaluation

Limit evaluation is the process of finding the value that a function approaches as the input approaches a particular point. It is a fundamental concept in calculus and allows us to explore behavior in functions around points of interest. In this exercise, we need to evaluate:

• \(\lim _{x \rightarrow 0} \frac{\tan^{-1} x-x}{8 x^{3}}\) which initially gives \(\frac{0}{0}\), leading to successive differentiation.
• The use of l'Hôpital's Rule involves the differentiation of both the numerator and the denominator, allowing us to evaluate limits of indeterminate forms.
• After differentiating, further simplification of the function helps in reevaluating the limit.

Through these procedures, we can often transform an indeterminate form into a determinate one, allowing the limit to be explicitly calculated. In our example, repeated application of l'Hôpital's Rule helps conclude that the limit evaluates to \(-\frac{1}{24}\).

Calculus Problem-Solving

Calculus problem-solving involves a set of strategies and methods to analyze and resolve problems involving calculus concepts such as limits, derivatives, and integrals. Strategies include:

• Identifying indeterminate forms, as they signal the appropriate scenarios for using techniques like l'Hôpital's Rule.
• Applying derivatives systematically to both the numerator and denominator in problems involving l'Hôpital's Rule.
• Simplifying expressions after differentiation to clearly see the structure of the problem.

In the exercise presented, problem-solving incorporated these techniques by identifying the indeterminate form, using l'Hôpital's Rule twice to differentiate and simplify the expression, and finally evaluating the limit. This multi-step process showcases the systematic approach often necessary in calculus to resolve seemingly complex mathematical challenges with precision and clarity.