Asymptotically Equal Real Functions/Examples

Examples of Asymptotically Equal Real Functions

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 $x \to 0$.