$$ \lim_{x\rarr\infty} $$$$ \sqrt{3x\text{ }} $$-2

Let's solve the limit:

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

### Step-by-Step Solution:

1. **Understand the behavior as $x \to \infty$:**  
   - As $x$ becomes very large, $\sqrt{3x}$ also becomes very large, because the square root function grows indefinitely for large $x$.
   - Subtracting 2 from a large number won’t affect the overall behavior much, but we need to calculate this properly.

2. **Extract common factors (Optional):**  
   - For large values of $x$, $\sqrt{3x}$ is dominant. However, directly taking the limit will suffice. We expect this expression to approach infinity because:

   \[
   \lim_{x \to \infty} \left( \sqrt{3x} - 2 \right) = \infty
   \]

### Conclusion:
The limit of $\sqrt{3x} - 2$ as $x \to \infty$ is $ \infty $. This means the expression grows unboundedly large as $x$ increases.

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**Related Questions:**
1. What is the limit of $\sqrt{3x + 5} - \sqrt{3x}$ as $x \to \infty$?
2. How do we handle limits with square roots and logarithms?
3. What happens if the expression is $\sqrt{3x - 1} - \sqrt{2x - 4}$ as $x \to \infty$?
4. How does L'Hôpital's rule apply to limits involving roots?
5. What is the general strategy for evaluating $ \lim_{x \to \infty} \sqrt{ax + b} $ for constants $a$ and $b$? 

**Tip:**  
When evaluating limits with square roots, it’s helpful to factor out dominant terms or multiply by conjugates to simplify tricky expressions.

The query submitted by the user: $$ \lim_{x\rarr\infty} \sqrt{3x} - 2 $$

Understand the limit of the expression sqrt(3x) - 2 as x approaches infinity. This problem involves core calculus concepts like limits and square root functions. A detailed step-by-step solution is provided for students in Grades 11-12.

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