Big-O Estimate for Real Function/Examples/10x Function at Infinity

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