Problem 85 - Calculus Early Transcendentals

Question

Use l'Hôpital's Rule to evaluate the following limits. \(\lim _{x \rightarrow 0} \frac{\tanh ^{-1} x}{\tan (\pi x / 2)}\)

Step-by-Step Solution

Answer

Question: Find the limit of the given function as x approaches 0: \(\lim_{x \rightarrow 0} \frac{\tanh ^{-1}(x)}{\tan(\pi x / 2)}\) Answer: \(\frac{2}{\pi}\)

Step 1: Identifying the conditions for applying l'Hôpital's Rule

We have the limit \(\lim_{x \rightarrow 0} \frac{\tanh ^{-1}(x)}{\tan(\pi x / 2)}\). Now we need to verify if the conditions for applying l'Hôpital's Rule are satisfied. As \(x \rightarrow 0\), we have the \(\tanh^{-1}(0) = 0\) and \(\tan(0) = 0\). So, our function is of the form 0/0 when \(x \rightarrow 0\).

Step 2: Differentiating the numerator and the denominator

Now, we will differentiate both the numerator and the denominator with respect to x. Numerator derivative: \(\frac{d}{dx}(\tanh^{-1}(x)) = \frac{1}{1 - x^2}\) Denominator derivative: \(\frac{d}{dx}(\tan(\pi x / 2)) = \frac{\pi}{2} \cdot \cos(\pi x / 2)^{-2}\)

Step 3: Apply l'Hôpital's Rule

Now, we apply l'Hôpital's Rule by taking the limit of the derivatives' ratio as x approaches 0. \(\lim_{x \rightarrow 0} \frac{\frac{d}{dx}(\tanh^{-1}(x))}{\frac{d}{dx}(\tan(\pi x / 2))} = \lim_{x \rightarrow 0} \frac{\frac{1}{1 - x^2}}{\frac{\pi}{2} \cdot \cos(\pi x / 2)^{-2}}\)

Step 4: Evaluate the limit

Now, let's substitute 0 for x in our new limit of derivatives. \(\lim_{x \rightarrow 0} \frac{\frac{1}{1 - 0^2}}{\frac{\pi}{2} \cdot \cos(\pi 0 / 2)^{-2}} = \frac{\frac{1}{1}}{\frac{\pi}{2} \cdot \cos(0)^{-2}} = \frac{2}{\pi}\) So, the final answer is \(\lim _{x \rightarrow 0} \frac{\tanh ^{-1} x}{\tan (\pi x / 2)} = \frac{2}{\pi}\).

Key Concepts

Limits

Limits are a fundamental concept in calculus used to describe the behavior of a function as its input approaches a particular value. When we say we are evaluating the limit of a function as \(x\) approaches a value, we're interested in what value the function itself approaches.

• The limit notation \(\lim_{x \rightarrow a} f(x)\) indicates what \(f(x)\) approaches as \(x\) gets closer to \(a\).
• Understanding limits is crucial for defining derivatives and integrals, as they provide a foundation for analyzing infinite processes.
• In the given exercise, we're looking at the limit as \(x\) approaches 0, which is essential for applying l'Hôpital's Rule.

To evaluate such a limit, checking the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) is key to applying l'Hôpital's Rule. This rule helps to take such indeterminate forms and turn them into a solvable derivatives-based problem.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but for a hyperbola instead of a circle. The hyperbolic tangent function, \(\tanh(x)\), and its inverse, \(\tanh^{-1}(x)\), are particularly important.

• Hyperbolic functions often arise in scenarios involving exponential growth and relate to many areas in applied mathematics and physics.
• The function \(\tanh^{-1}(x)\) provides the angle whose hyperbolic tangent is \(x\).
• It has a derivative of \(\frac{1}{1-x^2}\), which is used in step 2 of the solution.

Understanding the behavior of hyperbolic functions, especially their derivatives, is crucial in problems involving inverse hyperbolic components, as seen in evaluating the given limit problem.

Derivatives

Derivatives represent the rate at which a function is changing at any given point and are fundamental to calculus. They are the backbone of l'Hôpital's Rule.

• The derivative of a function \(f(x)\) at a point \(x\) is the slope of the tangent line to the curve at \(x\).
• Taking derivatives of both the numerator and denominator allows l'Hôpital's Rule to convert an indeterminate limit into a determinable limit.
• In the exercise, we differentiate \(\tanh^{-1}(x)\) and \(\tan(\pi x / 2)\) to address the 0/0 form, converting the problem using their derivatives.

Thus, derivatives are pivotal for understanding changes in functions and calculating limits that initially seem unsolvable.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent stem from studying angles and their relationships in triangles. In this exercise, the tangent function is particularly important.

• Trigonometric functions are periodic and are used to model cyclic patterns in nature, such as waves.
• The derivative of \(\tan(x)\), and specially \(\tan(\pi x / 2)\), requires the chain rule and gives \(\frac{\pi}{2} \cdot \cos(\pi x / 2)^{-2}\).
• Knowing the trigonometric derivatives helps in simplifying and solving complex limit problems through l'Hôpital's application.

Understanding the derivatives of trigonometric functions deepens comprehension of calculus techniques used to solve limits and differential equations efficiently.