Little-O Estimate for Real Function/Examples/x is Little-O of x^2 at Infinity

Example of Little-$\oo$ Estimate for Real Function

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = x$

Then:

$\map f x = \map \oo {x^2}$

as $x \to \infty$.

Let us consider the real function $g: \R \to \R$ defined as:

$\forall x \in \R: \map g x = x^2$

Then we have that:

$\ds \forall x \in \R_{\ne 0}: \, $

$\ds \dfrac {\map f x} {\map g x}$

$=$

$\ds \dfrac x {x^2}$

$\ds $

$=$

$\ds \dfrac 1 x$

$\ds \leadsto \ \ $

$\ds \lim_{x \mathop \to \infty} \dfrac {\map f x} {\map g x}$

$=$

$\ds \lim_{x \mathop \to \infty} \dfrac 1 x$

$\ds $

$=$

$\ds 0$

Hence the result.

$\blacksquare$