Saturday, 25 May 2019

real analysis - Find limntoinftyfrac(sqrt[n](7n+n)frac17)n7nn7





Find
limn(n(7n+n)17)n7nn7




I know that it can be done with using the squeeze theorem but I cannot find a proper upper bound limit


Answer



You may write
(n(7n+n)17)n7nn7=(7(1+n7n)1/n17)n7n(1n77n)=7n(1+O(17n)149)n7n(1n77n)=(1+O(17n)149)n(1n77n)=(4849)n(1+O(17n))n1n77n=(4849)n(1+O(n7n))1n77n(4849)n

and the desired limit is equal to 0.


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