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{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!