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We are tasked with computing the following limit:

\[
\lim_{x \to \infty} \left( 3x - \sqrt{9x^2 + 4x + 6} \right)
\]

### Step-by-Step Solution:

1. **Rewrite the expression:**
   We need to find the behavior of this expression as \( x \to \infty \). First, factor out \( x^2 \) inside the square root to handle the limit more easily:
   
   \[
   \lim_{x \to \infty} \left( 3x - \sqrt{9x^2 + 4x + 6} \right)
   = \lim_{x \to \infty} \left( 3x - \sqrt{x^2(9 + \frac{4}{x} + \frac{6}{x^2})} \right)
   \]

2. **Simplify the square root:**
   Since \( x^2 \) is factored out, the square root becomes:

   \[
   \sqrt{x^2(9 + \frac{4}{x} + \frac{6}{x^2})} = x\sqrt{9 + \frac{4}{x} + \frac{6}{x^2}}
   \]
   
   So, the limit expression simplifies to:

   \[
   \lim_{x \to \infty} \left( 3x - x \sqrt{9 + \frac{4}{x} + \frac{6}{x^2}} \right)
   \]

3. **Handle the limit as \( x \to \infty \):**
   As \( x \to \infty \), both \( \frac{4}{x} \) and \( \frac{6}{x^2} \) approach 0. Therefore, the expression inside the square root approaches:

   \[
   \sqrt{9 + 0 + 0} = 3
   \]

   So, the limit becomes:

   \[
   \lim_{x \to \infty} \left( 3x - 3x \right)
   \]

4. **Final calculation:**

   \[
   \lim_{x \to \infty} (3x - 3x) = \lim_{x \to \infty} 0 = 0
   \]

### Conclusion:
The limit of the given expression is:

\[
\boxed{0}
\]

Would you like a deeper explanation of any step? Here are five related questions to further explore:

1. How do you handle limits at infinity with square roots in general?
2. What techniques can be used to simplify complex expressions with limits?
3. Why do terms involving \( \frac{1}{x} \) vanish as \( x \to \infty \)?
4. How does factoring help in solving limits of rational or square root functions?
5. Can you explain how to apply L'Hopital's Rule for indeterminate forms?

**Tip:** Always factor expressions involving square roots to simplify the evaluation of limits at infinity!

Compute the following limit: lim(x→∞)(3x - √(9x^2 + 4x + 6))

Learn how to compute the limit of the expression 3x - √(9x^2 + 4x + 6) as x approaches infinity. This step-by-step solution covers simplification techniques and core limit concepts. Ideal for students studying calculus.

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