Assume f(x) goes to infinity as x tends to infinity and a,b>0. Find the f(x) that yields the lowest order for
as x tends to infinity. By order I mean Big O and Little o notation.
I can only solve it roughly:
My solution: We can say ln(1+f(x)) is approximately equal to ln(f(x)) as x goes to infinity. Then, I have to minimize the order of
Since for any c>0, y+c/y is miminized when y =sqrt(c), b+ln f(x)}=sqrt(ax) is the anwer. Equivalently, f(x)=e^(sqrt(ax)-b) and the lowest order for g(x) is 2 sqrt(ax).
Can you help me obtain a rigorous answer?