Definition:Asymptotic Equality/Real Functions

Definition

Estimate at infinity

Let $f$ and $g$ real functions defined on $\R$.

Then:

$f$ is asymptotically equal to $g$

if and only if:

$\dfrac {\map f x} {\map g x} \to 1$ as $x \to +\infty$.

That is, the larger the $x$, the closer $f$ gets (relatively) to $g$.

Point Estimate

Let $f$ and $g$ real functions defined on a punctured neighborhood of $x_0$.

Then:

$f$ is asymptotically equal to $g$ at $x_0$

if and only if:

$\dfrac {\map f x} {\map g x} \to 1$ as $x \to x_0$.

That is, the closer $x$ gets to $x_0$, the closer $f$ gets (relatively) to $g$.

Examples

Example: $x$ and $x + 1$

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 $x \to +\infty$.