equivalent functions?

Question

I have this two functions in ($1<x<50$)

$y = -1/x$

and

$ y = \frac{x - \sqrt{x^2+4}}{2} $

why this are very similar ?

Answer 1

The reason is that $$\sqrt{4+x^2}=x+\frac{2}{x}-\frac{2}{x^3}+O((\frac{1}{x})^5)$$

Plugging this into your equation $y=-\frac{1}{x}+\frac{1}{x^3}+O((\frac{1}{x})^5)$.