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

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Example of Limit of Real Function


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


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


By definition of the limit of a real function:

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

if and only if:

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

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

Let $\delta = \epsilon$.

Then we have:

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

Hence the result.