Question

determine the following limit.
\\( \lim_{\theta \to \infty} \frac{\sin 4\theta}{14\theta} \\)

select the correct choice and, if necessary, fill in the answer box to complete your choice.

\\( \bigcirc \\) a. \\( \lim_{\theta \to \infty} \frac{\sin 4\theta}{14\theta} = \square \\) (simplify your answer.)
\\( \bigcirc \\) b. the limit does not exist and is neither \\( -\infty \\) nor \\( \infty \\).

Explanation:

Step1: Analyze the range of sine function

The sine function \( \sin x \) has a range of \( [-1, 1] \), so for \( \sin 4\theta \), we have \( -1 \leq \sin 4\theta \leq 1 \).

Step2: Analyze the denominator as \( \theta \to \infty \)

As \( \theta \to \infty \), the denominator \( 14\theta \) also approaches \( \infty \).

Step3: Analyze the limit of the fraction

We have a fraction \( \frac{\sin 4\theta}{14\theta} \) where the numerator is bounded between -1 and 1, and the denominator approaches \( \infty \). By the Squeeze Theorem (or the property of limits of bounded functions over infinite functions), if \( -1 \leq f(\theta) \leq 1 \) and \( g(\theta) \to \infty \), then \( \lim_{\theta \to \infty} \frac{f(\theta)}{g(\theta)} = 0 \). So \( \lim_{\theta \to \infty} \frac{\sin 4\theta}{14\theta} = 0 \).

Answer:

A. \( \lim\limits_{\theta \to \infty} \frac{\sin 4\theta}{14\theta} = \boxed{0} \)