Problem 32 - Thomas Calculus

Question

Find the limits. Are the functions continuous at the point being approached? $$\lim _{t \rightarrow 0} \sin \left(\frac{\pi}{2} \cos (\tan t)\right)$$

Step-by-Step Solution

Answer

The limit is 1 and the function is continuous at \( t = 0 \).

Step 1: Evaluate the Inner Function

The inner function that needs to be evaluated as \( t \) approaches 0 is \( \tan t \). As \( t \rightarrow 0 \), \( \tan t \rightarrow 0 \).

Step 2: Evaluate the Cosine

We now substitute the limit of the inner function into the cosine function: \( \cos(\tan t) \). Since \( \tan t \rightarrow 0 \), \( \cos(\tan t) \rightarrow \cos(0) = 1 \).

Step 3: Evaluate the Argument of the Sine Function

The next expression to evaluate is \( \frac{\pi}{2} \cos(\tan t) \) as \( t \rightarrow 0 \). Substituting from Step 2, this expression becomes \( \frac{\pi}{2} \cdot 1 = \frac{\pi}{2} \).

Step 4: Evaluate the Sine Function

Substitute into the sine function: \( \sin\left(\frac{\pi}{2}\right) \). Since the sine of \( \frac{\pi}{2} \) is 1, \( \sin\left(\frac{\pi}{2}\right) = 1 \).

Step 5: Conclusion on Continuity

Since all component functions (\( \tan t \), \( \cos x \), and \( \sin x \)) involved are continuous at the point being approached, the overall function is continuous at \( t = 0 \).

Key Concepts

Trigonometric Functions

Trigonometric functions, like sine, cosine, and tangent, are fundamental in mathematics. These functions relate to angles and sides of triangles and can describe waves and oscillations. Each of these functions behaves in a predictable pattern:

• **Sine** (\( \sin \, \)): This function oscillates between -1 and 1. It starts at 0 when the angle is 0 and peaks at 1 when the angle reaches \( \frac{\pi}{2} \).
• **Cosine** (\( \cos \, \)): Similar to sine, cosine also ranges from -1 to 1. It starts at 1 when the angle is 0 and decreases to 0 at \( \frac{\pi}{2} \).
• **Tangent** (\( \tan \)): The tangent function is the ratio of sine to cosine. Unlike sine and cosine, its range is all real numbers and it has vertical asymptotes at odd multiples of \( \frac{\pi}{2} \).

In this specific problem, we needed to calculate the limit of a combination of these functions. This involves substituting through each function carefully until reaching the final value of the limit.

Evaluating Limits

Limits are used to understand the behavior of functions as input values approach a specific point. When evaluating limits, especially for composite functions, it's important to break down the steps:

• **Inner Function**: Start by determining the limit of the innermost function, which is known as evaluating inside-out.
• **Substitute**: Once the inner limit is known, you substitute it into the next outer function.
• **Simplify**: Continue substituting until the entire expression is simplified and the limit is clear.

In the exercise provided, we first evaluated \( \tan t \) as \( t \rightarrow 0 \), which simplified to 0. Next, \( \cos(0) \) was found, which equals 1. Finally, \( \frac{\pi}{2} \cos(0) \) simplifies to \( \frac{\pi}{2} \), and then \( \sin(\frac{\pi}{2}) \) leads to the limit of 1. This step-by-step decomposition leads to a clear understanding of the process of evaluating limits.

Continuity of Functions

A function is continuous at a point if it has no breaks, jumps, or holes at that point. There are a few characteristics to consider:

• **No Discontinuities**: The function must not have any points where it "jumps" or is undefined.
• **Limit Equals Function Value**: The limit of the function as it approaches the point must equal the function's value at that point.
• **Overall Continuity**: If a function is composed of continuous operations like addition, subtraction, etc., from continuous functions, it remains continuous.

In our example, \( \tan t \), \( \cos x \), and \( \sin x \) are continuous functions. When these are combined into a composite function, they remain continuous at \( t = 0 \). Hence, the overall function in the problem is continuous at \( t = 0 \) as each individual function involved is continuous at that point.