<s>[INST] Finding big O of a function How do I find Big O of function which are polynomial fractions
$$f(x) = \frac {x^4 + x^2 + 1}{x^3 + 1}$$
The same question is posted here (Finding Big-O with Fractions) but i dont understand the explanation on how from the following we concluded that it is order x
$$f(x) = \frac {x^4 + x^2 + 1}{x^3 + 1} 
= \frac{(x^4 + x^2 +1)/x^3}{(x^3 + 1)/x^3}
= \frac{x + \frac{1}{x} + \frac{1}{x^3}}{1 + \frac{1}{x^3}}$$
and you can prove directly from this that $f(x) = O(x)$.
Also, what are the witness variables for the above proof?
 [/INST] For every $x\geqslant1$, $x\leqslant f(x)\leqslant x+1$. Can you conclude?
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