Tuesday, 30 December 2014

real analysis - Limit limlimitsnto+inftyesqrtncdotleft(1frac1sqrtnright)n



limn+ en(11n)n



Answers are:



A)0




B)1



C)e



D)e



E)1e



I tried working on the second part to get it in a better form. In the end I got en. Returning at the beginning with this new form it would be:

limn+ enen

which would eventually turn into limn+ e0
so the answer would be B) 1 but it's not. The answer is E)1e but I can't figure it out why.


Answer



Consider A=en(11n)n

Take logarithms of both sides log(A)=n+nlog(11n)
Now, use the fact that for small values of x, log(1x)=xx22+O(x3). Replace x by 1n which makes log(A)=n+n(1n12n+)
Expand and simplify.



I am sure that you can take from here.


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