Substitution in Big-O Estimate/Real Analysis

Theorem

Let $f$ and $g$ be real-valued or complex-valued functions defined on a neighborhood of $+ \infty$ in $\R$.

Let $f = \map \OO g$, where $\OO$ denotes big-O notation.

Let $h$ be a real-valued defined on a neighborhood of $+ \infty$ in $\R$.

Let $\ds \lim_{x \mathop \to +\infty} \map h x = +\infty$.

Then:

$f \circ h = \map \OO {g \circ h}$ as $x \to +\infty$.

Proof

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