Question

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 3} \tan \left(\frac{\pi x}{4}\right) $$

Short answer

The limit of the function \(\tan(\frac{\pi x}{4})\) as \(x\) approaches 3 is \(-1\).

Step-by-step solution

Understand the Problem

The task is about finding the limit of the function \(\tan(\frac{\pi x}{4})\) as \(x\) approaches 3. The function is continuous, and no indeterminate form arises from direct substitution, so we can directly substitute the value \(x=3\) into the function.

Substitute the Value

Substitute \(x=3\) into \(\tan(\frac{\pi x}{4})\), giving \(\tan(\frac{3 \pi}{4})\).

Find the Tan Value

Knowing that \(\tan(\frac{3\pi}{4})\) is -1, the limit of our function is \(-1\).

Key concepts

Trigonometric Limits

When dealing with trigonometric limits, we address the behavior of trigonometric functions as the input approaches a particular value. This involves analyzing functions like sine, cosine, and tangent as the input, usually denoted as \( x \), approaches a target value. The goal is to determine what value the function approaches as \( x \) nears this specific point. Trigonometric limit problems often require simplification, especially when facing indeterminate forms or discontinuities. For the function \( \tan(\frac{\pi x}{4}) \), the limit as \( x \) approaches 3 is straightforward because the function is continuous around the target value.
This straightforwardness means we can often use direct substitution to find the limit, given the function does not involve division by zero or other complexities.

Continuous Functions

Continuous functions are those where small changes in the input result in small changes in the output, without abrupt jumps or breaks. This smooth behavior allows us to use direct substitution methods when evaluating limits.
For continuous trigonometric functions like \( \tan(\frac{\pi x}{4}) \), evaluating the limit often involves directly inserting the target input value into the function. Continuous functions benefit from predictability: as \( x \) gets closer to our target value, the function's output similarly approaches the limit.

• Helps avoid complicated limit simplifications or algebraic manipulations.
• Ensures seamless substitution without concerns of undefined behaviors such as division by zero.

As a result, understanding continuity can simplify the overall process of finding trigonometric limits.

Direct Substitution

Direct substitution is a method used to find limits by directly replacing the variable with its approaching value. This technique works perfectly for continuous functions where no indeterminate forms, such as \( \frac{0}{0} \), arise. In the exercise \( \lim_{x \rightarrow 3} \tan(\frac{\pi x}{4}) \), we can safely substitute \( x = 3 \) because the function is continuous at this point and no interruptions occur.
The steps for direct substitution are simple:

• Identify the limit expression, in this case, \( \tan(\frac{\pi x}{4}) \) as \( x \) approaches 3.
• Substitute \( x = 3 \) directly into the expression.
• Calculate the resulting function value, which here is \( \tan(\frac{3\pi}{4}) = -1 \).

This confirms our limit, allowing a quick solution without excessive algebra.

Tangent Function

The tangent function, \( \tan(\theta) \), relates to the sine and cosine functions through the identity \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). Known for its periodic nature, the tangent function repeats values every \( \pi \) radians. At angles where the cosine is zero, the tangent function becomes undefined due to division by zero.
In this exercise, we're finding \( \tan(\frac{3\pi}{4}) \) as part of evaluating the limit \( \lim_{x \rightarrow 3} \tan(\frac{\pi x}{4}) \). The angle \( \frac{3\pi}{4} \) corresponds to the second quadrant of the unit circle, where cosine is negative and sine is positive, leading to a tangent value of \(-1\). Understanding these properties allows efficient calculation of limits involving tangent, ensuring precise results when direct substitution reveals a straightforward value.