Given the information: $\lim_{x \to 2} f(x) = 4, \quad \lim_{x \to 2} g(x) = 3, \quad \text{and} \quad \lim_{x \to 2} h(x) = 0,$ we need to find $\lim_{x \to 2} \left( \frac{f(x)h(x) + g(x)h(x)}{g(x)} \right).$

Step-by-Step Solution

1. Simplify the Expression Inside the Limit: $\frac{f(x)h(x) + g(x)h(x)}{g(x)} = \frac{h(x)(f(x) + g(x))}{g(x)}.$

   This can be rewritten as: $h(x) \cdot \frac{f(x) + g(x)}{g(x)}.$

2. Apply Limit Laws: Since $\lim_{x \to 2} h(x) = 0$, we have: $\lim_{x \to 2} \left[ h(x) \cdot \frac{f(x) + g(x)}{g(x)} \right] = \left( \lim_{x \to 2} h(x) \right) \cdot \left( \lim_{x \to 2} \frac{f(x) + g(x)}{g(x)} \right).$

3. Evaluate Each Limit Separately:

   • $\lim_{x \to 2} h(x) = 0$.
   • $\lim_{x \to 2} \frac{f(x) + g(x)}{g(x)} = \frac{\lim_{x \to 2} (f(x) + g(x))}{\lim_{x \to 2} g(x)}$ since $\lim_{x \to 2} g(x) \neq 0$.
   • Evaluate $\lim_{x \to 2} (f(x) + g(x)) = \lim_{x \to 2} f(x) + \lim_{x \to 2} g(x) = 4 + 3 = 7$.
   • Thus, $\lim_{x \to 2} \frac{f(x) + g(x)}{g(x)} = \frac{7}{3}$.

4. Combine the Results: $\left( \lim_{x \to 2} h(x) \right) \cdot \left( \lim_{x \to 2} \frac{f(x) + g(x)}{g(x)} \right) = 0 \cdot \frac{7}{3} = 0.$

Final Answer:

The limit is $0$. Therefore, the correct option is:

• 0

Would you like further details or have any other questions?

Related Questions:

1. What are the laws of limits used in this problem?
2. How would the solution change if $\lim_{x \to 2} h(x)$ were not equal to zero?
3. Why is it necessary to ensure $\lim_{x \to 2} g(x) \neq 0$ when applying limit division?
4. Can limits be distributed over addition and multiplication?
5. How do you handle indeterminate forms in limit problems?

Tip:

Always break down the limit into simpler parts and apply limit laws step-by-step to avoid confusion, especially when dealing with products and quotients.