Evaluate the following limit: `lim_(n->oo){sqrt(x+1)-sqrt(x)}sqrt(x+2)`

Explanation

To evaluate the limit \( L = \lim_{x \to \infty} \left( \sqrt{x+1} - \sqrt{x} \right) \sqrt{x+2} \), we can follow these steps:

### Step 1: Simplify the expression
We start with the expression \( \sqrt{x+1} - \sqrt{x} \). To simplify this, we can multiply and divide by the conjugate:

\[ \sqrt{x+1} - \sqrt{x} = \frac{(\sqrt{x+1} - \sqrt{x})(\sqrt{x+1} + \sqrt{x})}{\sqrt{x+1} + \sqrt{x}} = \frac{(x+1) - x}{\sqrt{x+1} + \sqrt{x}} = \frac{1}{\sqrt{x+1} + \sqrt{x}} \]
...