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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.

Given lim(x→2) f(x) = 4, lim(x→2) g(x) = 3, and lim(x→2) h(x) = 0, calculate lim(x→2) [ (f(x)h(x) + g(x)h(x)) / g(x) ]

Solve a limit problem involving the functions f(x), g(x), and h(x) with given limits at x = 2. Learn how to apply limit laws to simplify the expression and find the result.

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