Determine whether $x^3$ is $O(g(x))$ for certain functions $g(x)$.

Question

a) $g(x) = x^2$

b) $g(x) = x^3$

c) $g(x) = x^2 + x^3$

d) $g(x) = x^2 + x^4$

e) $g(x) = 3^x$

f) $g(x) = (x^3)/2$

Do you guys have any ideas? Thanks!

Answer 1

As $x\to\infty $ :

Well , for polynomials you should just look at the term with highest degree.

Positive coefficients are negligible.

And $a^x$ always dominate polynomials in the long run for $a>1$..

So , for your question :all possible except a...