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: Understand the Problem

The problem is asking us to find the limit as \( \theta \) approaches 0 for the function \( \cos \left(\frac{\pi \theta}{\sin \theta}\right) \). This requires analyzing the behavior of the expression inside the cosine function as \( \theta \) approaches 0.

Step 2: Simplify the Expression Inside Cosine

Inside the cosine function, we have the expression \( \frac{\pi \theta}{\sin \theta} \). As \( \theta \to 0 \), we know from the standard trigonometric identity that \( \sin \theta \sim \theta \). Thus, \( \frac{\theta}{\sin \theta} \to 1 \). So, \( \frac{\pi \theta}{\sin \theta} \) approaches \( \pi \cdot 1 = \pi \).

Step 3: Evaluate the Limit of the Simplified Expression

Now, substituting the limit value of the expression inside the cosine function, we evaluate the limit: \( \lim_{\theta \to 0} \cos \left(\pi \cdot 1\right) = \cos(\pi) \).

Step 4: Compute the Final Limit

Knowing that \( \cos(\pi) = -1 \), the limit of the cosine function evaluated at the simplified expression as \( \theta \) approaches 0 is \(-1\).

Key Concepts

trigonometric limits

Trigonometric limits are essential concepts in calculus that help us understand the behavior of trigonometric functions as a variable approaches a particular value. When dealing with limits involving trigonometric functions, it's crucial to understand certain fundamental limits and properties. Trigonometric functions like sine and cosine exhibit unique behaviors due to their periodic nature and symmetry. When evaluating trigonometric limits, particularly as a variable approaches zero, we often rely on standard trigonometric identities and approximations like

• \( \sin \theta \sim \theta \) as \( \theta \) approaches 0, making it easier to simplify expressions.
• \( \cos \theta \sim 1 \) for small \( \theta \).
• Utilizing these approximations can simplify the evaluation process significantly.

Understanding these properties allows us to tackle more complex expressions and find their limits. Recognizing patterns and approximations helps make the calculations both simpler and more accurate.

cosine function

The cosine function, denoted as \( \cos(x) \), is one of the basic trigonometric functions. It describes the x-coordinate of a point on the unit circle as the angle \( x \) varies. Here are some crucial properties:

• The cosine function is periodic with a period of \( 2\pi \). This mean \( \cos(x + 2\pi) = \cos(x) \).
• It has a range from -1 to 1, making it suitable for modeling oscillating phenomena.
• The cosine function is even, meaning \( \cos(-x) = \cos(x) \), which is useful when simplifying expressions involving cosine.
• At specific angles, the cosine function takes on simple values: \( \cos(0) = 1 \), \( \cos(\pi/2) = 0 \), and \( \cos(\pi) = -1 \).

In our specific exercise, the cosine function was used to evaluate the simplified expression \( \cos(\pi) \), which equals -1. Understanding these properties enables evaluating expressions involving the cosine function with confidence and precision.

limit evaluation techniques

Evaluating limits is a core skill in calculus, and various techniques can aid in this process. Choosing the right approach depends on the form of the expression and the behavior of the function involved. Some common limit evaluation techniques include:

• **Direct Substitution**: When possible, substitute the value directly into the expression to find the limit.
• **L'Hôpital's Rule**: Applicable when you have an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). You differentiate the numerator and denominator separately to resolve the indeterminacy.
• **Trigonometric Identities**: As seen in our exercise, using identities like \( \sin \theta \sim \theta \) can help simplify complex trigonometric limits.
• **Algebraic Manipulation**: Sometimes, rewriting or factoring an expression can make a limit more apparent.

In our original exercise, we applied trigonometric identities to simplify the inner expression. This allowed us to evaluate the cosine at a known angle, demonstrating how critical the right technique is for solving limit problems efficiently.