A little change goes a long way

$\begin{aligned} L & = \lim_{x \to \infty} \sqrt{2x(x+1)} - x & \small \color{#3D99F6} \text{Multiplied by }\frac {\sqrt{2x(x+1)} + x}{\sqrt{2x(x+1)} + x} \\ & = \lim_{x \to \infty} \frac {2x(x+1) - x^2}{\sqrt{2x(x+1)} + x} \\ & = \lim_{x \to \infty} \frac {x^2+2x}{\sqrt{2x^2+2x} + x} & \small \color{#3D99F6} \text{Divide up and down by }x \\ & = \lim_{x \to \infty} \frac {x+2}{\sqrt{2+\frac 2x} + 1} \\ & = \boxed{\infty} \end{aligned}$

$\lim_{x \to \infty} \sqrt{2x(x+1)} - x = \lim_{x \to \infty} \frac{ \left( \sqrt{2x(x+1)} - x \right) \left( \sqrt{2x(x+1)} + x \right)}{ \sqrt{2x(x+1)} + x } = \lim_{x \to \infty} \frac{2x(x+1) - x^2}{\sqrt{2x(x+1)}+x} = \lim_{x \to \infty} \frac{O(x^2)}{O(x)} = \infty$