Find the limit.
$\lim_{x\rarr-\pi}\frac{\sin x}{x}$

• A. $-\frac{1}{\pi}$
• B. $0$
• C. $\pi$
• D. Does not exist

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the limit of the function \( \frac{\sin x}{x} \) as \( x \) approaches \( -\pi \).
2. Step 2: Recall the limit properties. The limit of a function \( \frac{f(x)}{g(x)} \) as \( x \) approaches a point can be found if both \( f(x) \) and \( g(x) \) approach a finite value and \( g(x) \neq 0 \).
3. Step 3: Evaluate the behavior of \( \sin x \) and \( x \) as \( x \) approaches \( -\pi \). Note that \( \sin(-\pi) = 0 \) and \( x \) approaches \( -\pi \).
4. Step 4: Consider the form \( \frac{0}{-\pi} \). Since \( \sin(-\pi) = 0 \), the numerator approaches 0, and the denominator approaches \( -\pi \), which is a non-zero constant.
5. Step 5: Conclude that the limit exists and is equal to \( \frac{0}{-\pi} \).