Definition:Asymptotic Equality/Real Functions/Infinity

Definition

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 $g$.

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 $x \to +\infty$.

Example: $x^2 + 1$ and $x^2$

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 $x \to +\infty$.

Example: $\sin x$ and $x$

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

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

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

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

Then:

$f \sim g$

as $x \to 0$.

Sources

• 1979: G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers (5th ed.) ... (previous) ... (next): $\text I$: The Series of Primes: $1.6$ Some notations