The problem you've uploaded asks for the following limit evaluation:

$\lim_{x \to 0} \frac{\sin x}{3x^2 + 2}$

Let's break it down:

Step 1: Substitute $x = 0$

We start by attempting direct substitution of $x = 0$ into the expression:

$\frac{\sin(0)}{3(0)^2 + 2} = \frac{0}{0 + 2} = \frac{0}{2} = 0$

Step 2: Interpret the limit

Since substituting $x = 0$ results in a straightforward answer of 0, there's no need for advanced techniques like L'Hopital's Rule.

Therefore, the correct value of the limit is:

$\lim_{x \to 0} \frac{\sin x}{3x^2 + 2} = 0$

However, the options provided don't include this result, which suggests there might be a typo or missing details. Double-checking the question or choices may help clarify. If you're certain of the options, let me know.

Would you like more detailed steps or clarification on any part? Here's a tip: When solving limits, always check if direct substitution leads to indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. If not, simple evaluation can suffice!

Here are 5 related questions you might find useful:

1. What happens when you apply L'Hopital's Rule to limits?
2. How do you handle indeterminate forms in limits like $\frac{0}{0}$?
3. What is the general behavior of the function $\frac{\sin x}{x}$ as $x \to 0$?
4. How do you find limits of rational functions as $x \to 0$?
5. What role does continuity play in evaluating limits?

Tip: When evaluating limits, check if the numerator and denominator approach 0, indicating the potential for using L'Hopital's Rule.