Question

find the limit using \\(\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1\\).
\\(\lim_{x \to 0} \frac{x \csc 3x}{\cos 7x}\\)

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

\\(\bigcirc\\) a. \\(\lim_{x \to 0} \frac{x \csc 3x}{\cos 7x} = \square\\) (simplify your answer.)
\\(\bigcirc\\) b. the limit does not exist.

Explanation:

Step1: Recall csc definition

$\csc 3x = \frac{1}{\sin 3x}$, so rewrite the limit:
$\lim_{x\to 0} \frac{x \cdot \frac{1}{\sin 3x}}{\cos 7x} = \lim_{x\to 0} \frac{x}{\sin 3x \cos 7x}$

Step2: Manipulate for standard limit

Multiply numerator and denominator by 3:
$\lim_{x\to 0} \frac{3x}{3\sin 3x \cos 7x} = \lim_{x\to 0} \frac{1}{3 \cdot \frac{\sin 3x}{3x} \cdot \cos 7x}$

Step3: Apply limit rules

Use $\lim_{\theta\to 0} \frac{\sin \theta}{\theta} = 1$ (let $\theta = 3x$) and $\lim_{x\to 0} \cos 7x = \cos 0 = 1$:
$\frac{1}{3 \cdot 1 \cdot 1} = \frac{1}{3}$

Answer:

A. $\lim\limits_{x\to 0} \frac{x \csc 3x}{\cos 7x} = \frac{1}{3}$