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The problem you're working on involves calculating the limit:

\[
\lim_{x \to \infty} 2x \left( \sqrt{9 + \frac{10}{x}} - 3 \right)
\]

### Step-by-step solution:

1. **Consider the limit behavior** as \( x \to \infty \). The expression inside the square root can be simplified:
   
   \[
   \sqrt{9 + \frac{10}{x}}
   \]
   As \( x \to \infty \), \( \frac{10}{x} \to 0 \), so:
   
   \[
   \sqrt{9 + \frac{10}{x}} \to \sqrt{9} = 3
   \]
   Thus, the expression becomes close to \( 3 - 3 = 0 \), but to handle the limit more carefully, we need to expand the square root.

2. **Apply a binomial expansion** for the square root for large \( x \):
   
   \[
   \sqrt{9 + \frac{10}{x}} = 3 + \frac{1}{2 \cdot 3} \cdot \frac{10}{x} + O\left(\frac{1}{x^2}\right)
   \]
   This simplifies to:
   
   \[
   \sqrt{9 + \frac{10}{x}} \approx 3 + \frac{5}{3x}
   \]

3. **Substitute back into the original expression**:
   
   \[
   2x \left( \left(3 + \frac{5}{3x}\right) - 3 \right) = 2x \cdot \frac{5}{3x}
   \]
   Simplifying this gives:
   
   \[
   2x \cdot \frac{5}{3x} = \frac{10}{3}
   \]

4. **Conclusion**:
   
   The value of the limit is \( \frac{10}{3} \), and expressed with three decimal places, the result is:

   \[
   3.333
   \]

### Final Answer:
\[
\boxed{3.333}
\]

Let me know if you would like further details or clarification!

---

Here are some related questions for practice:
1. What is the limit of \( \lim_{x \to \infty} x \left( \sqrt{1 + \frac{1}{x}} - 1 \right) \)?
2. How does the binomial expansion help in simplifying limits involving square roots?
3. What happens when you calculate \( \lim_{x \to \infty} \left( \sqrt{x^2 + 1} - x \right) \)?
4. Can you solve \( \lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{x} \)?
5. How can L'Hopital's Rule be applied to similar limit problems?

**Tip**: When dealing with limits involving square roots and infinity, the binomial expansion can be a very effective tool.

Tentukan Nilai dari Hasil dari lim x->∞ 2x (√(9 + 10/x) - 3).

Solve the limit lim x->∞ 2x (√(9 + 10/x) - 3) using binomial expansion and algebraic techniques. This problem covers key concepts in limits and expansions, suitable for high school students in Grades 11-12.

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