Problem 50 - University Calculus: Early Transcendentals

Question

Find the limits. $$\lim _{\theta \rightarrow \pi / 4} \frac{\tan \theta-1}{\theta-\frac{\pi}{4}}$$

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Identify the Limit Type

First, notice that as \( \theta \to \frac{\pi}{4} \), both the numerator \( \tan \theta - 1 \) and the denominator \( \theta - \frac{\pi}{4} \) approach 0, which indicates this is an indeterminate form \( \frac{0}{0} \). This suggests that techniques like L'Hôpital's Rule or algebraic manipulation are appropriate.

Step 2: Apply L'Hôpital's Rule

Since the expression \( \frac{\tan \theta - 1}{\theta - \frac{\pi}{4}} \) is in the indeterminate form \( \frac{0}{0} \), apply L'Hôpital's Rule. This rule stipulates that the limit of a \( \frac{0}{0} \) form can be found as follows: Compute the derivative of the numerator and the derivative of the denominator.

Step 3: Find the Derivatives

Compute the derivative of the numerator: The derivative of \( \tan \theta \) is \( \sec^2 \theta \). Thus, the derivative of \( \tan \theta - 1 \) is \( \sec^2 \theta \).Compute the derivative of the denominator: The derivative of \( \theta - \frac{\pi}{4} \) is simply 1.

Step 4: Evaluate the Limit with Derivatives

Substitute the derivatives into L'Hôpital's Rule to find the limit:\[ \lim_{\theta \to \frac{\pi}{4}} \frac{\sec^2 \theta}{1} = \sec^2 \left( \frac{\pi}{4} \right) \]Recall that \( \sec \theta = \frac{1}{\cos \theta} \). At \( \theta = \frac{\pi}{4} \), \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\), so \( \sec \frac{\pi}{4} = \frac{1}{\cos \frac{\pi}{4}} = \sqrt{2} \). Hence \( \sec^2 \frac{\pi}{4} = 2 \).

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus for resolving indeterminate forms like \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \). When you have these forms, direct substitution won't work. Instead, L'Hôpital's Rule provides a method to find the limit through differentiation. Here's how it works:

• First, check that the limit leads to an indeterminate form.
• Then, differentiate both the numerator and the denominator separately.
• Finally, compute the limit of these new derivatives.

In this exercise, the original expression \( \frac{\tan \theta - 1}{\theta - \frac{\pi}{4}} \) is \( \frac{0}{0} \) as \( \theta \to \frac{\pi}{4} \). By applying L'Hôpital's Rule, we calculate the derivatives: \( \sec^2 \theta \) for the numerator and 1 for the denominator. Substituting these into the limit gives you a much easier expression to evaluate.

Trigonometric Limits

Trigonometric limits can often appear tricky, but understanding them is essential for handling calculus problems involving trigonometric functions. To compute limits involving trigonometric functions like \( \tan x \), \( \sin x \), and \( \cos x \), you often rely on fundamental trigonometric identities and the behavior of these functions near specific points.

• Knowledge of trigonometric identities helps simplify expressions.
• Recognize special angles, such as \( \frac{\pi}{4} \), to find exact trigonometric values.
• Trigonometric limits often require transforming into simpler forms or using limit laws and theorems.

In our exercise, evaluating \( \sec^2\left( \frac{\pi}{4} \right) \) involves recognizing that \( \theta = \frac{\pi}{4} \) corresponds to an angle where both sine and cosine have specific known values. Specifically, \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), leading to \( \sec \frac{\pi}{4} = \sqrt{2} \) and consequently \( \sec^2 \frac{\pi}{4} = 2 \). Understanding these limits is crucial for efficiently evaluating trigonometric limits.

Indeterminate Forms

Indeterminate forms arise in calculus when substituting a limit value results in nonsensical expressions like \( \frac{0}{0} \) or \( \infty - \infty \). These forms require special techniques to resolve, as they don't give definite information about the behavior of the function around the point.

• The most common indeterminate form is \( \frac{0}{0} \), often addressed by L'Hôpital's Rule.
• Others include multiples like \( 0 \cdot \infty \) or \( 1^\infty \), which need algebraic manipulation or specific limit theorems.
• Recognizing indeterminate forms is key to determining the correct approach. This might involve derivatives, series expansions, or trigonometric identities.

In this problem, both \( \tan \theta - 1 \) and \( \theta - \frac{\pi}{4} \) tend to zero as \( \theta \to \frac{\pi}{4} \). Thus, we find the expression \( \frac{\tan \theta - 1}{\theta - \frac{\pi}{4}} \) doesn’t provide useful information directly. Applying L'Hôpital’s Rule turns this into \( \frac{\sec^2 \theta}{1} \), a clear path to assessing the limit. Understanding indeterminate forms guides you to use the right technique for tough limits.