Problem 24 - Precalculus

Question

Use a table of values to evaluate each function as \(x\) approaches the value indicated. If the function seems to approach a limiting value, write the relationship in words and using the limit notation. $$q(x)=\tan x+3 \sin 2 x ; x \rightarrow \frac{\pi}{4}$$

Step-by-Step Solution

Answer

\( \lim_{x \to \frac{\pi}{4}} (\tan x + 3 \sin 2x) = 4. \)

Step 1: Understand the function components

The given function is composed of two main trigonometric components: \( \tan x \) and \( 3 \sin 2x \). As \( x \to \frac{\pi}{4} \), we need to evaluate both parts separately and then sum their results.

Step 2: Evaluate \( \tan x \) as \( x \to \frac{\pi}{4} \)

\( \tan x = \frac{\sin x}{\cos x} \). As \( x \) approaches \( \frac{\pi}{4} \), both \( \sin x \) and \( \cos x \) approach \( \frac{\sqrt{2}}{2} \). Thus, \( \tan \frac{\pi}{4} = \frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}} = 1 \).

Step 3: Evaluate \( 3 \sin 2x \) as \( x \to \frac{\pi}{4} \)

The function \( 3 \sin 2x \) is evaluated by considering \( \sin 2x \). When \( x = \frac{\pi}{4} \), \( 2x = \frac{\pi}{2} \). Therefore, \( \sin \frac{\pi}{2} = 1 \). So, \( 3 \sin 2 \cdot \frac{\pi}{4} = 3 \cdot 1 = 3 \).

Step 4: Sum the evaluated components

The function \( q(x) = \tan x + 3 \sin 2x \) is composed of the two parts we evaluated: \( 1 \) for \( \tan x \) and \( 3 \) for \( 3 \sin 2x \). Summing these values gives \( 1 + 3 = 4 \).

Step 5: State the limiting value using limit notation

Based on the evaluations of \( \tan x \) and \( 3 \sin 2x \), as \( x \to \frac{\pi}{4} \), \( q(x) \to 4 \). Therefore, we conclude that \( \lim_{x \to \frac{\pi}{4}} (\tan x + 3 \sin 2x) = 4 \).

Key Concepts

Trigonometric Functions

Trigonometric functions are a crucial part of mathematics, especially when dealing with angles and periodic phenomena. In the context of our exercise, we have two such trigonometric expressions: \( \tan x \) and \( 3 \sin 2x \).

**Understanding \( \tan x \)**
The tangent function, \( \tan x \), is defined as the ratio of sine to cosine: \( \tan x = \frac{\sin x}{\cos x} \). As angles increase, so does the tangent value, due to the periodic and asymmetrical nature of the sine and cosine functions. For example, at \( x = \frac{\pi}{4} \), both sine and cosine are equal to \( \frac{\sqrt{2}}{2} \), and hence \( \tan \frac{\pi}{4} \) results in \( 1 \).

**Understanding \( 3 \sin 2x \)**
The function \( 3 \sin 2x \) involves a sine function that is doubled, \( \sin 2x \), meaning its period is halved. This doubling results in a faster cycle of sine's oscillation. For \( x = \frac{\pi}{4} \), \( \sin(2 \times \frac{\pi}{4}) = \sin \frac{\pi}{2} = 1 \). Scaling by 3 gives us \( 3 \times 1 = 3 \).

These functions are not only essential for calculating angles in simple geometric problems but also in more complex scenarios like wave motion and signal processing.

Limit Notation

Limit notation is a powerful tool in calculus used to describe the behavior of functions as they approach a certain point. In this exercise, we explore the behavior of a function, \( q(x) = \tan x + 3 \sin 2x \), as \( x \) approaches \( \frac{\pi}{4} \).

To express this in limit notation, we write it as \( \lim_{x \to \frac{\pi}{4}} (\tan x + 3 \sin 2x) \). The limit helps us understand what happens to \( q(x) \) near \( x = \frac{\pi}{4} \) without having to substitute the exact value directly into the function. This is especially useful when dealing with indeterminate forms or discontinuities.

In our exercise, both components \( \tan x \) and \( 3 \sin 2x \) have clearly defined values as \( x \to \frac{\pi}{4} \), allowing us to conclude that the entire function approaches the value of 4 at this point. Limits thus provide a concise way to express this kind of information.

Evaluation of Functions

Evaluating a function means finding its output for given inputs. In our exercise, we had to evaluate \( q(x) = \tan x + 3 \sin 2x \) as \( x \) approached \( \frac{\pi}{4} \).

The evaluation involves calculating each part of the function separately before combining them for the total output.

• First, compute the \( \tan x \) at \( x = \frac{\pi}{4} \), which simplifies to 1.
• Then, calculate \( 3 \sin 2x \) for the same \( x \), which simplifies to 3.

Finally, sum these results to get the total: \( 1 + 3 = 4 \). This summation provides the final value that the function approaches, confirming that the limit of \( q(x) \) as \( x \) tends to \( \frac{\pi}{4} \) is indeed 4. Evaluating functions like this can be a critical skill in understanding real-world phenomena and creating mathematical models.