I've spent the last hours trying to understand Big O and little O notations in depth. Usually the limit of the quotient method works but here, I'm dealing with integrals which seems to make things harder.
integral(dt/ln(t),t=2..sqrt(x))=O(sqrt(x)) when x->+∞
integral(dt/(ln(t))^2,t=sqrt(x)..x)=o(integral(dt/ln(t),t=sqrt(x)..x) when x->+∞
I'd really appreciate some tips or the general method on proving these two assertions. Sorry for not knowing Latex, I'll get to it soon ^^
Thanks for any help!
Yes that is what I've been able to do so far but I'm not sure this allows me to verify the first assertion. I think it has to do with the
∃x0>0 ∃M>0 ∀x>x0 abs(f(x))<M*abs(g(x)) => f(x)=O(g(x)) but I'm not even sure it is a proper definition and why the condition x->+∞ is useful here. Thanks for your answer by the way!
Ok I think I get it, I can just create the quotients with those inequalities (since everything is positive and ≠ 0) and then just use the Squeeze theorem.
If this is all I need to do, please just confirm so that I can put the little [SOLVED] Thank you very much for your help CaptainBlack!