Asymptotically Equal Real Functions/Examples/sin x and x

Example of Asymptotically Equal Real Functions

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 $x \to 0$.

Proof

\(\ds \dfrac {\map f x} {\map g x}\)

\(=\)

\(\ds \dfrac {\sin x} x\)

\(\ds \leadsto \ \ \)

\(\ds \lim_{x \mathop \to 0} \dfrac {\map f x} {\map g x}\)

\(=\)

\(\ds 1\)

Limit of Sine of X over X at Zero

$\blacksquare$

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): asymptotic: 2.