Problem 12 - Calculus with Concepts in Calculus

Question

Use the results of this section to evaluate the limit. $$ \lim _{t \rightarrow 3 \pi / 2} \sin \left(\frac{\pi}{2} \sin t\right) $$

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Identify the Type of Limit and Substitution

The given limit is \( \lim_{t \rightarrow \frac{3\pi}{2}} \sin\left(\frac{\pi}{2} \sin t\right) \). We need to investigate the behavior of \( \sin\left(\frac{\pi}{2} \sin t\right) \) as \( t \) approaches \( \frac{3\pi}{2} \). Start by recognizing that \( \sin t \) will be evaluated at \( t = \frac{3\pi}{2} \).

Step 2: Evaluate the Sine Function at the Limit Point

Recall that \( \sin \left( \frac{3\pi}{2} \right) = -1 \). Therefore, as \( t \) approaches \( \frac{3\pi}{2} \), \( \sin t \) will approach \(-1\). This means the expression inside the sine function becomes \( \sin\left(\frac{\pi}{2} (-1) \right) = \sin\left(-\frac{\pi}{2}\right). \)

Step 3: Simplify and Compute the Inner Sine

Using the property of the sine function, \( \sin(-x) = -\sin(x) \), we find \( \sin\left(-\frac{\pi}{2}\right) = -\sin\left(\frac{\pi}{2}\right) = -1 \).

Step 4: Conclusion of the Limit Evaluation

Therefore, as \( t \) approaches \( \frac{3\pi}{2} \), \( \sin\left(\frac{\pi}{2} \sin t\right) \) approaches \(-1\). This evaluation tells us the final result of the limit.

Key Concepts

Understanding the Sine Function

The sine function is one of the basic trigonometric functions, typically denoted as \( \sin \theta \). It is defined for all real numbers and represents the y-coordinate of a unit circle centered at the origin. When you hear about the sine function, it's important to recall how it cycles through values, repeating every \( 2\pi \) radians.
The key points of this cycle are:

• At \( \theta = 0 \), \( \sin(\theta) = 0 \).
• At \( \theta = \pi/2 \), \( \sin(\theta) = 1 \).
• At \( \theta = \pi \), \( \sin(\theta) = 0 \).
• At \( \theta = 3\pi/2 \), \( \sin(\theta) = -1 \).

As \( t \) approached \( \frac{3\pi}{2} \) in the exercise, it resulted in \( \sin(t) = -1 \). This knowledge helps determine how sine functions interact within other functions, crucial for evaluating limits.

Exploring Trigonometric Limits

Evaluating trigonometric limits often involves understanding how trigonometric functions behave as they approach certain angles. Trigonometric limits can be tricky but become manageable with practice and familiarity with fundamental trigonometric values.
Let's break this down with a few notes:

• As \( t \) approaches a specific value, consider the known values of \( \sin, \cos, \) and other trigonometric functions at these points.
• Focus on the behavior of functions as they approach multiples of \( \pi \).
• Remember common angle values such as \( \pi/2, \pi, \) and \( 3\pi/2 \).

In the exercise, as \( t \) was evaluated approaching \( 3\pi/2 \), the knowledge of \( \sin(t) \) directly affected the inner expression leading to limit evaluation.

The Limit Evaluation Process

The limit evaluation process is a fundamental technique in calculus, especially when dealing with complicated expressions such as those involving trigonometric functions. When approaching a limit evaluation problem, follow these steps:

• Identify the Type of Limit: Recognize the form of the function that you're taking the limit of.
• Simplify the Expression: Use algebraic or trigonometric identities to simplify the function when possible.
• Substitute the Limit Point: See if direct substitution helps or if the expression results in an indeterminate form.
• Apply Trigonometric Identities: Use identities like \( \sin(-x) = -\sin(x) \) or others to further simplify expressions.
• Re-Evaluate to Confirm: Once simplified, re-assess the expression as the variable approaches the given point to make sure the evaluation is correct.

In the provided step-by-step solution, after confirming that \( \sin(t) ≈ -1 \), further substitutions were done to arrive at the final limit value of \(-1\). Understanding this process is vital to mastering calculus limits.