Substitution in Big-O Estimate/Real Analysis

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Theorem

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

Let $f = O(g)$, where $O$ denotes big-O notation.

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

Let $\lim_{x\to+\infty} h(x) = +\infty$.


Then $f\circ h = O(g\circ h)$ as $x\to+\infty$.


Proof