1. **State the problem:** Find the limit $$\lim_{x \to 2} (\sqrt{x+1} - x)$$. 2. **Recall the limit and algebraic manipulation technique:** When direct substitution leads to an indeterminate form, we can multiply by the conjugate to simplify. 3. **Direct substitution:** Substitute $x=2$: $$\sqrt{2+1} - 2 = \sqrt{3} - 2$$ which is a defined number, so the limit is simply $\sqrt{3} - 2$. 4. **Conclusion:** Since the expression is continuous at $x=2$, the limit is: $$\boxed{\sqrt{3} - 2}$$. No further simplification is needed.