Identity to simplify $x \times \left( 1-\sqrt{1-\frac{2}{x}} \right)$

Question

I notice that with $x \to \infty$, $x \times \left( 1-\sqrt{1-\frac{2}{x}} \right) \approx 1$.

Is there any well-known identity to prove this?

Answer 1

$$ \frac{1}{1+\sqrt{1-\frac{2}{x}}}=\left(1-\sqrt{1-\frac{2}{x}}\right)\times \frac{1}{1-(1-\frac{2}{x})}=\frac{x}{2}\left(1-\sqrt{1-\frac{2}{x}}\right) $$

Answer 2

$f(x)=x\left(1-\sqrt{1-\dfrac{2}{x}}\right) $

$x\left(1-\sqrt{1-\dfrac{2}{x}}\right) =x-\sqrt{\dfrac{x-2}{x}} x=x-\sqrt{\dfrac{x^2 (x-2)}{x}}=$

$=x-\sqrt{x^2-2 x}=\dfrac{\left(x-\sqrt{x^2-2 x}\right) \left(x+\sqrt{x^2-2 x}\right)}{x+\sqrt{x^2-2 x}}=\dfrac{x^2-\left(x^2-2 x\right)}{\sqrt{x^2-2 x}+x}=$

$=\dfrac{2 x}{\sqrt{x^2 \left(1-\dfrac{2}{x}\right)}+x}=\dfrac{2 x}{x\sqrt{1-\dfrac{2}{x}} +x}\to \dfrac{2x}{2x}=1,\text {as }x\to\infty$