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?

$$\dfrac{x^4+x^2+1}{x^3+1} = \dfrac{(x^2+x+1)(x^2-x+1)}{(x+1)(x^2-x+1)} = \dfrac{x^2+x+1}{x+1} = x + \dfrac1{x+1}$$
Now you should be able to conclude what you want.
