Limit of Real Function/Examples/x times Sine of Reciprocal of x at 0
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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:
\(\ds 0 < \size x\) | \(<\) | \(\ds \delta\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {x \map \sin {\dfrac 1 x} }\) | \(<\) | \(\ds \delta\) | 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.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.3$: Limits of functions: Examples $1.3.4$