Asymptotically Equal Real Functions/Examples/x^2+1 and x^2

Example of Asymptotically Equal Real Functions

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

$\forall x \in \R: \map f x = x^2 + 1$

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

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

Then:

$f \sim g$

as $x \to +\infty$.

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

$=$

$\ds \dfrac {x^2 + 1} {x^2}$

$\ds $

$=$

$\ds 1 + \frac 1 {x^2}$

dividing top and bottom by $x^2$

$\ds \leadsto \ \ $

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

$=$

$\ds 1$

$\blacksquare$