Saturday, August 26, 2017

calculus - Proving that limntoinftyleft(1+frac1f(n)right)g(n)=1



I want to prove that limn(1+1f(n))g(n)=1 if f(n) grows faster than g(n) for n and limnf(n)=+=limng(n).



It is quite easy to see that if f=g the limit is e, but I can't find a good strategy to solve this problem.


Answer



We can use that
(1+1f(n))g(n)=[(1+1f(n))f(n)]g(n)f(n)


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