Problem 44 - Thomas Calculus

Question

Find the limits in Exercises \(39-44\) $$ \lim _{\theta \rightarrow 0} \cos \left(\frac{\pi \theta}{\sin \theta}\right) $$

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Recall Basic Trigonometric Limit

In this limit problem, as \( \theta \to 0 \), we should remember that one of the key trigonometric limits used frequently is \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). This property will be useful in simplifying the expression.

Step 2: Simplify the Inner Expression

The expression inside the cosine function is \( \frac{\pi \theta}{\sin \theta} \). To evaluate the limit, we rearrange this as \( \pi \cdot \frac{\theta}{\sin \theta} \). We know from the trigonometric limit that as \( \theta \to 0 \), \( \frac{\theta}{\sin \theta} \to 1 \). Thus, the expression simplifies to \( \pi \cdot 1 = \pi \).

Step 3: Evaluate the Limit of the Entire Expression

Now, having simplified \( \frac{\pi \theta}{\sin \theta} \) to \( \pi \), we can evaluate the outer function: \( \lim_{\theta \to 0} \cos(\pi) \). The cosine of \( \pi \) is \( -1 \). Therefore, the limit is \( -1 \).

Key Concepts

Trigonometric Limits

Trigonometric limits are an essential topic in calculus, especially when dealing with expressions that involve sine, cosine, or other trigonometric functions. A common trigonometric limit used is \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). This limit is the foundation for evaluating more complex trigonometric expressions. When approaching zero, these limits are a tool to simplify or transform complex trigonometric expressions, making them easier to handle in calculations. This concept is used frequently in both pure math and real-world applications involving periodic phenomena.

Limit Evaluation

Limit evaluation is the process of finding the value that a function approaches as the input approaches a certain point. In the context of the given exercise, it involves using known limit properties and simplifications to determine the behavior of the function as \( \theta \) approaches zero. Often, limits can initially appear complex but become manageable by applying trigonometric identities and transformations to simplify the terms. For instance:

• Identifying common limits like \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \).
• Rewriting expressions to utilize these limits effectively.

The goal is to break down the expression so each part complies with known limits, leading to an accurate evaluation.

Cosine Function

The cosine function, \( \cos(x) \), is a fundamental trigonometric function that describes the adjacent side over hypotenuse in a right triangle for angle \( x \). It is periodic with a period of \( 2\pi \), oscillating between -1 and 1. For exercises involving limits, the behavior of the cosine function is crucial, especially when the argument of cosine approaches known values like \( \pi \) or \( 0 \). For instance, \( \cos(\pi) = -1 \). This property can immediately provide the limit of expressions that contain the cosine function when other parts have been simplified.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. They allow transformations and simplifications of expressions. In the given exercise, the identity used was based on the limit: \( \frac{\theta}{\sin \theta} \to 1 \) as \( \theta \to 0 \). Knowing these identities helps in rewriting and reducing expressions to a simpler form. Some basic identities include:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)
• \( 1 - \cos^2 \theta = \sin^2 \theta \)

These can be instrumental when approaching limits as they facilitate easier manipulation of expressions to reach a conclusive result.