Show $f(x) = (x^4+x^2+1)/(x^3+1) $ is $O(x)$ How would I find the witnesses $C$ and $k$ such that $f(x)$ is $O(x)$?
What I tried was $$(x^4+x^2+1)/(x^3+1) ≤ (x^4+x^4+x^4)/(x^3+x^3) = (3/2)x $$
for values $x>1$. $C = 3/2, k = 1$
Is this right?

We have
$$
f(x)= \frac{x^4 + x^2 +1 }{x^3 +1} \leq  C x,
$$
for $x \geq \sqrt{\frac{C}{3(C-1)}}$ and $C>1$.
