Proving the asymptotic relationship between $(lg\cdot n)^{0.5}$ and $lg\cdot (n^{0.5})$?

Question

Say $f(n) = (lg\cdot n)^{0.5}$

and $g(n) = lg\cdot (n^{0.5})$

It would appear that $f(n) = O(g(n))$ for $n \gt 55$ correct? How do I go about proving the the relationship for this?

Answer 1

You don't need specific values for $n$ necessarily to show that $f(n) = O(g(n))$. All you need is that the limit $f(n)/g(n)$ goes to a non-negative constant as $n \to \infty$ (even 0) or that $f(n)/g(n)$ is bounded above by some non-negative constant for $n$ large enough.