Little-O Estimate for Real Function/Examples/Sine Function at Infinity
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Example of Little-$\oo$ Estimate for Real Function
Let $f: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map f x = \sin x$
Then:
- $\map f x = \map \oo x$
as $x \to \infty$.
Proof
Let us consider the real function $g: \R \to \R$ defined as:
- $\forall x \in \R: \map g x = x$
Then we have that:
\(\ds \forall x \in \R_{> 0}: \, \) | \(\ds \dfrac {\map f x} {\map g x}\) | \(=\) | \(\ds \dfrac {\sin x} x\) | |||||||||||
\(\ds \) | \(\le\) | \(\ds \dfrac 1 x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lim_{x \mathop \to \infty} \dfrac {\map f x} {\map g x}\) | \(=\) | \(\ds \lim_{x \mathop \to \infty} \dfrac {\sin x} x\) | |||||||||||
\(\ds \) | \(\le\) | \(\ds \lim_{x \mathop \to \infty} \dfrac 1 x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
Hence the result.
$\blacksquare$
Sources
- 1979: G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers (5th ed.) ... (previous) ... (next): $\text I$: The Series of Primes: $1.6$ Some notations