We are tasked with solving the following limit:

$\lim_{x \to \infty} x\left( \sqrt{x^2 + 2x} - 2\sqrt{x^2 + x + x} \right)$

Step-by-Step Solution:

1. Simplifying the square root terms:

For large $x$, we can factor $x^2$ out of each square root to simplify them.

• In $\sqrt{x^2 + 2x}$, factor $x^2$ out: $\sqrt{x^2 + 2x} = \sqrt{x^2(1 + \frac{2}{x})} = x\sqrt{1 + \frac{2}{x}}$

• Similarly, for $2\sqrt{x^2 + 2x}$: $2\sqrt{x^2 + 2x} = 2x\sqrt{1 + \frac{2}{x}}$

2. Applying the limit:

As $x \to \infty$, the terms $\frac{2}{x} \to 0$, so we can approximate:

• $\sqrt{1 + \frac{2}{x}} \to 1$
• $2\sqrt{1 + \frac{2}{x}} \to 2$

Thus, for large $x$, the expression inside the limit becomes approximately:

[ x \left( x(1 + \frac{2}{x}) - 2x\left( ) = --- Placeholder