1. **State the problem:** Find the limit as $x$ approaches infinity of the expression $$\sqrt{x+1} - \sqrt{x+2}.$$\n\n2. **Recall the formula and technique:** When dealing with limits involving differences of square roots, a useful technique is to multiply and divide by the conjugate to simplify the expression. The conjugate of $$\sqrt{a} - \sqrt{b}$$ is $$\sqrt{a} + \sqrt{b}.$$\n\n3. **Apply the conjugate:** Multiply numerator and denominator by $$\sqrt{x+1} + \sqrt{x+2}$$ to get\n$$\lim_{x \to \infty} \frac{(\sqrt{x+1} - \sqrt{x+2})(\sqrt{x+1} + \sqrt{x+2})}{\sqrt{x+1} + \sqrt{x+2}} = \lim_{x \to \infty} \frac{(x+1) - (x+2)}{\sqrt{x+1} + \sqrt{x+2}}.$$\n\n4. **Simplify numerator:**\n$$ (x+1) - (x+2) = x + 1 - x - 2 = -1.$$\nSo the expression becomes\n$$\lim_{x \to \infty} \frac{-1}{\sqrt{x+1} + \sqrt{x+2}}.$$\n\n5. **Analyze the denominator as $x \to \infty$:** Both $$\sqrt{x+1}$$ and $$\sqrt{x+2}$$ behave like $$\sqrt{x}$$ for large $x$, so the denominator grows without bound.\n\n6. **Evaluate the limit:** Since the numerator is constant (-1) and the denominator grows without bound, the whole fraction approaches 0 from the negative side.\n\n**Final answer:**\n$$\boxed{0}.$$