# Limit of Real Function/Examples/x times Sine of Reciprocal of x at 0

## Example of Limit of Real Function

Let:

$\map f x = x \map \sin {\dfrac 1 x}$

Then:

$\ds \lim_{x \mathop \to 0} \map f x = 0$

## Proof

By definition of the limit of a real function:

$\ds \lim_{x \mathop \to 0} \map f x = A$
$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size x < \delta \implies \size {\map f x - A} < \epsilon$

Let $\epsilon \in \R_{>0}$ be chosen arbitrarily.

Let $\delta = \epsilon$.

Then we have:

 $\ds 0 < \size x$ $<$ $\ds \delta$ $\ds \leadstoandfrom \ \$ $\ds \size {x \map \sin {\dfrac 1 x} }$ $<$ $\ds \delta$ because $\map \sin {\dfrac 1 x} \le 1$ for $x \ne 0$ $\ds \leadstoandfrom \ \$ $\ds \size {x \map \sin {\dfrac 1 x} }$ $<$ $\ds \epsilon$ $\ds \leadstoandfrom \ \$ $\ds \size {\map f x - 0}$ $<$ $\ds \epsilon$

Hence the result.

$\blacksquare$