Big O notation and limits

Question

I'm wanting to take the $\lim_{x\to \infty} \frac {O(1)}{x^s}$, where $O(1)$ is Big O notation and $s>1$. I can see that it will be zero but I'm wanting to do it somewhat rigorously. Can I take the Lim inside the Big O or should I consider some other approach with sequences etc.

Answer 1

Rigorously, you might say something like this (keep in mind that $f(x)=O(1)$ if $x>N$ then $f(x)\leq C$ for some constant $C$).

$$\lim_{x\to\infty} \frac{f(x)}{x^{s}} \leq \lim_{x\to\infty}\frac{C}{x^{s}}=C\lim_{x\to\infty}\frac{1}{x^{s}} = C\cdot 0 = 0$$

Answer 2

Use the definition of $O(1)$: bounded if $x$ is large enough. Btw, $$\frac{O(1)}{x^s}=O\Bigl(\frac 1{x^s}\Bigr).$$