Why is $x(\sqrt{x^2+1} - x )$ approaching $1/2$ when $x$ becomes infinite? Why is $$\lim_{x\to \infty}  x\left( (x^2+1)^{\frac{1}{2}} - x\right)  = \frac{1}{2}?$$
What is the right way to simplify this?
My only idea is: $$x((x²+1)^{\frac{1}{2}} - x) > x(x^{2^{0.5}}) - x^2 = 0$$
But 0 is too imprecise.

Hint: $\sqrt{x^2 + 1} - x$ can be written as $\sqrt{x^2 + 1} - \sqrt{x^2}$.
Generally, if you have an expression involving $\sqrt a - \sqrt b$ then it
is often useful to "expand" the expression with $\sqrt a + \sqrt b$:
$$
  x \, (\sqrt{x^2 + 1} - x) = \frac {x \,(\sqrt{x^2 + 1} - x)(\sqrt{x^2 + 1} + x)} {\sqrt{x^2 + 1} + x}
$$
