Friday, 23 February 2018

calculus - The limit as n approaches infinity of n(a1/n1)



I need to know how to calculate this without using l'hospitals rule:



limit as x approaches infinity of: x(a1/x1)



I saw that the answer is log(a), but I want to know how they got it.

The book implies that I should be able to find it by just using algebraic manipulation and substitution.


Answer



METHOD 1:



limxx(a1/x1)=limx0+ax1x=limx0+exloga1x=limx0+(1+(loga)x+O(x2))1x=limx0+(loga+O(x))=loga






METHOD 2:



Another way to do this is to substitute y=ax in (1). Then



limxx(a1/x1)=limx0+ax1x=limy1+y1logy/loga=logalimy1+y1logy



Noting that for y>1, y1ylogyy1. Then,



1y1logyy



and the squeeze theorem does the rest!


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