Problem 76 - Precalculus Mathematics for Calculus

Question

A function \(f\) is given. (a) Is \(f\) even, odd, or neither? (b) Find the \(x\) -intercepts of the graph of \(f\) (c) Graph \(f\) in an appropriate viewing rectangle. (d) Describe the behavior of the function as \(x \rightarrow \pm \infty\). (e) Notice that \(f(x)\) is not defined when \(x=0 .\) What happens as \(x\) approaches \(0 ?\) $$f(x)=\frac{\sin 4 x}{2 x}$$

Step-by-Step Solution

Answer

(a) Even; (b) x-intercepts are \(x = \frac{n\pi}{4}, n \neq 0\); (c) Graph shows oscillation; (d) As \(x \rightarrow \pm \infty\), \(f(x)\rightarrow 0\); (e) As \(x \rightarrow 0\), \(f(x) \rightarrow 2\).

Step 1: Determine Even, Odd, or Neither

To determine if the function is even, odd, or neither, calculate \(f(-x)\) and compare it to \(f(x)\) and \(-f(x)\). The function is even if \(f(-x) = f(x)\), odd if \(f(-x) = -f(x)\), and neither otherwise. For \(f(x) = \frac{\sin 4x}{2x}\), \[f(-x) = \frac{\sin(-4x)}{2(-x)} = \frac{-\sin(4x)}{-2x} = \frac{\sin(4x)}{2x} = f(x)\]Hence, \(f\) is even.

Step 2: Find the X-intercepts

The x-intercepts of a function occur where \(f(x) = 0\). Solve \(\frac{\sin 4x}{2x} = 0\) for \(x\). The sine function is zero when its argument is an integer multiple of \(\pi\), i.e., \[4x = n\pi \, \Rightarrow \, x = \frac{n\pi}{4}, \, n \in \mathbb{Z}\]Thus, the x-intercepts are at \(x = \frac{n\pi}{4}\) for \(n \in \mathbb{Z\setminus}\{0\}\) since the function is not defined at \(x=0\).

Step 3: Graph the Function

Graph \(f(x) = \frac{\sin 4x}{2x}\) using a graphing tool or calculator. Choose a viewing rectangle that includes several periods of the sine function and avoids \(x = 0\) where the function is undefined. A possible range: \(-2\pi \leq x \leq 2\pi\), \(-\frac{1}{2} \leq y \leq \frac{1}{2}\). Observe the oscillation and symmetry.

Step 4: Behavior as \(x \rightarrow \pm \infty\)

Consider the limit of \(f(x)\) as \(x\) approaches \(\pm \infty\):\[\lim_{x \to \pm\infty} \frac{\sin 4x}{2x}.\]The sine function oscillates between -1 and 1, so \(f(x)\) behaves like \(\frac{y}{x}\) where \(-1 \leq y \leq 1\). Therefore, \[\lim_{x \to \pm\infty} f(x) = 0.\]

Step 5: Behavior as \(x\) Approaches 0

Examine the behavior of \(f(x)\) as \(x \to 0\). The function is not defined at \(x = 0\), but can use L'Hôpital's Rule as \(x \to 0\), since it forms \(\frac{0}{0}\):\[\lim_{x \to 0} \frac{\sin 4x}{2x} = \lim_{x \to 0} \frac{4\cos 4x}{2} = 2\cos 0 = 2.\]Thus, near \(x = 0\), \(f(x)\) approaches \(2\).

Key Concepts

Even and Odd Functions

To determine if functions are even or odd, we explore their symmetrical properties across the y-axis. If a function is even, it satisfies the condition: \( f(-x) = f(x) \). In contrast, an odd function satisfies: \( f(-x) = -f(x) \). If neither condition is met, the function is neither even nor odd.
For the function \( f(x) = \frac{\sin 4x}{2x} \), we compute \( f(-x) \) and find:

• \( f(-x) = \frac{\sin(-4x)}{2(-x)} = \frac{-\sin 4x}{-2x} = \frac{\sin 4x}{2x} = f(x) \).

This confirms that the function satisfies the even function condition. Even functions typically exhibit symmetry about the y-axis, providing a visual clue when graphing.

X-intercepts

X-intercepts are the points where a graph of a function crosses the x-axis. This occurs whenever \( f(x) = 0 \). Solving \( \frac{\sin 4x}{2x} = 0 \) allows us to find these intercepts.
The expression \( \sin 4x = 0 \) is zero when \( 4x \) equals an integer multiple of \( \pi \). Therefore:

• \( 4x = n\pi \)
• \( x = \frac{n\pi}{4} \)

Where \( n \) is any integer excluding zero because the function is undefined at \( x = 0 \). Understanding x-intercepts is essential because they indicate where the graph touches or crosses the x-axis. Some intercepts repeat periodically depending on the function's nature.

Limits at Infinity

Exploring the behavior of functions as \( x \) approaches infinity (\( \infty \)) or negative infinity (\( -\infty \)) is crucial to understanding asymptotic behavior. For the function \( f(x) = \frac{\sin 4x}{2x} \):

• The sine function oscillates between -1 and 1.
• Thus \( \frac{\sin 4x}{2x} \) behaves like \( \frac{y}{x} \), where \( -1 \leq y \leq 1 \).

This means that as \( x \to \pm \infty \), the function approaches zero:

• \( \lim_{x \to \pm \infty} \frac{\sin 4x}{2x} = 0 \).

Understanding limits at infinity helps predict how functions behave along extensive input ranges, offering insights into trends and asymptotes in the graph.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool for evaluating limits that result in indeterminate forms, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When you encounter such limits, L'Hôpital's Rule allows you to differentiate the numerator and the denominator separately and then recalculate the limit.
For \( f(x) = \frac{\sin 4x}{2x} \) as \( x \to 0 \), direct substitution yields \( \frac{0}{0} \), an indeterminate form. Applying L'Hôpital's Rule:

• Differentiate the numerator: \( \frac{d}{dx} (\sin 4x) = 4 \cos 4x \).
• Differentiate the denominator: \( \frac{d}{dx} (2x) = 2 \).

Thus, \( \lim_{x \to 0} \frac{\sin 4x}{2x} = \lim_{x \to 0} \frac{4 \cos 4x}{2} = 2 \cdot \cos 0 = 2 \).
This approach reveals that \( f(x) \) approaches 2 near \( x = 0 \), giving deeper insight into local behavior of the function near undefined points.