Big-O Estimate for Real Function/Examples/10x Function at Infinity
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Example of Big-$\OO$ Estimate for Real Function
Let $f: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map f x = 10 x$
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:
- $\forall x \in \R_{>0}: \size {\map f x } < 11 \cdot \size x$
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