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

Example of Asymptotically Equal Real Functions
Let $f: \R \to \R$ be the real function defined as:
 * $\forall x \in \R: \map f x = x + 1$

Let $g: \R \to \R$ be the real function defined as:
 * $\forall x \in \R: \map g x = x$

Then:
 * $f \sim g$

as $x \to +\infty$.

Proof
Since:
 * $\dfrac {\map f x} {\map g x} = 1 + \dfrac 1 x$

we have:
 * $\ds \lim _{x \mathop \to +\infty} \dfrac {\map f x} {\map g x} = 1$