Problem 22 - Calculus Early Transcendentals

Question

Evaluate the following limits using \(l\) Hópital's Rule. $$\operatorname{lim}_{\rightarrow 0} \frac{\tan 4 z}{\tan 7 z}$$

Step-by-Step Solution

Answer

Answer: The limit of \(\frac{\tan 4z}{\tan 7z}\) as \(z\) approaches \(0\) is \(\frac{4}{7}\).

Step 1: Find the derivatives of the functions in the numerator and denominator

First, we need to find the derivatives of both \(\tan 4z\) and \(\tan 7z\) with respect to \(z\). We use the chain rule: if \(u(z) = 4z\) and \(v(z) = 7z\), then \(\frac{d}{dz} \tan u(z) = \frac{d}{dz}(\tan 4z) = \sec^2(4z) \cdot \frac{d}{dz}(4z)\) and similarly for \(\tan 7z\). $$\frac{d}{dz}(\tan 4z) = \sec^2(4z) \cdot 4$$ $$\frac{d}{dz}(\tan 7z) = \sec^2(7z) \cdot 7$$

Step 2: Evaluate the limit using L'Hôpital's Rule

Now we'll apply L'Hôpital's Rule to find the limit: $$\lim_{\rightarrow z \to 0} \frac{\tan 4z}{\tan 7z} = \lim_{\rightarrow z \to 0} \frac{\sec^2(4z) \cdot 4}{\sec^2(7z) \cdot 7}$$ Simplify the expression inside the limit: $$\lim_{\rightarrow z \to 0} \frac{4 \sec^2(4z)}{7 \sec^2(7z)}$$

Step 3: Evaluate the limit

Now we can evaluate the limit, since we know that \(\sec 0 = 1\): $$\lim_{\rightarrow z \to 0} \frac{4 \sec^2(4z)}{7 \sec^2(7z)} = \frac{4 \sec^2(0)}{7 \sec^2(0)} = \frac{4}{7}$$

Step 4: Write the final answer

After evaluating the limit using L'Hôpital's Rule, we conclude that: $$\lim_{\rightarrow z \to 0} \frac{\tan 4z}{\tan 7z} = \frac{4}{7}$$

Key Concepts

Understanding Limits in Mathematics

In calculus, limits help us understand the behavior of functions as they approach certain points or infinity. They are essential for defining derivatives and integrals. When evaluating limits, especially for indeterminate forms (like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\)), L'Hôpital's Rule becomes extremely useful.
Here’s how it applies:

• Check if your limit results in an indeterminate form.
• If so, take the derivative of the numerator and the derivative of the denominator separately.
• Re-evaluate the limit using these derivatives.

This transforms a tricky problem into a solvable one, helping us to find the limit accurately.

The Role of Calculus in Derivatives

Calculus is the branch of mathematics that deals with the concept of change. A central theme in calculus is finding the derivative of a function.
Derivatives represent the rate at which a function changes at a given point, providing insights into the function's behavior.
When applying L'Hôpital's Rule, derivatives play a key part:

• You need to differentiate the numerator and the denominator of the function independently.
• For example, the derivative of \(\tan 4z\) is \(4 \sec^2(4z)\), found using the chain rule.
• The chain rule helps calculate the derivative of composite functions by multiplying the derivative of the inside function.

This process simplifies evaluating limits by transforming indeterminate forms into simpler expressions.

Working with Trigonometric Functions

Trigonometric functions, like \(\tan\), \(\sec\), \(\sin\), and \(\cos\), are fundamental in calculus and understanding angles and periodic phenomena.
They often appear in limit problems due to their wave-like properties.
For example, in our exercise involving \(\tan 4z\) and \(\tan 7z\):

• Knowing that \(\tan(z)\) is the ratio of \(\sin(z)\) to \(\cos(z)\) can be helpful.
• Derivatives of trigonometric functions are crucial — for instance, the derivative of \(\tan(z)\) is \(\sec^2(z)\).
• These derivatives allow us to apply L'Hôpital's Rule effectively by simplifying complex expressions.

Understanding these functions and their derivatives is essential for solving problems in both pure and applied mathematics.