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The inequality given is:

\[
x^2 + 5x \leq x f(x) \leq 3x + \sin(2x)
\]
We are asked to find:

\[
\lim_{x \to 0} f(x)
\]

### Step-by-Step Solution:

1. **Consider the lower bound:**

   From \(x^2 + 5x \leq x f(x)\), divide both sides by \(x\) (for \(x \neq 0\)):

   \[
   x + 5 \leq f(x)
   \]

   Taking the limit as \(x \to 0\):

   \[
   \lim_{x \to 0} (x + 5) = 5
   \]

   Therefore, we have:

   \[
   \lim_{x \to 0} f(x) \geq 5
   \]

2. **Consider the upper bound:**

   From \(x f(x) \leq 3x + \sin(2x)\), divide both sides by \(x\) (for \(x \neq 0\)):

   \[
   f(x) \leq 3 + \frac{\sin(2x)}{x}
   \]

   We know that:

   \[
   \lim_{x \to 0} \frac{\sin(2x)}{x} = 2
   \]

   Therefore, as \(x \to 0\):

   \[
   f(x) \leq 3 + 2 = 5
   \]

   Hence:

   \[
   \lim_{x \to 0} f(x) \leq 5
   \]

3. **Conclusion:**

   Since \(5 \leq \lim_{x \to 0} f(x) \leq 5\), it follows that:

   \[
   \lim_{x \to 0} f(x) = 5
   \]

Thus, the correct answer is **b) 5**.

Would you like more details on any step, or do you have any other questions?

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Here are five questions to expand your understanding:

1. What are the common techniques for solving limits in inequalities?
2. How does the Squeeze Theorem apply in this context?
3. Why do we divide by \(x\) in the inequality, and what happens when \(x = 0\)?
4. How can we analyze limits of trigonometric functions like \(\sin(2x)\)?
5. How would the solution change if the inequality bounds were different?

**Tip:** Always check for continuity and use the Squeeze Theorem when faced with inequalities in limit problems.

Find lim(x→0) f(x) where x^2 + 5x ≤ x*f(x) ≤ 3x + sin(2x).

Learn how to find lim(x→0) f(x) given the inequality x^2 + 5x ≤ x*f(x) ≤ 3x + sin(2x) using the Squeeze Theorem. This advanced-level problem integrates trigonometric limits and inequalities, ideal for students in Grades 11-12.

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