Monday, February 25, 2019

real analysis - How can I find limnrightarrowinfty(1+fracxn)sqrtn?




How can I find limn(1+xn)n?




I know limn(1+xn)n=exp(x) but I don't know how can I put the definition in this particular limit.




I know then, that limn(1+xn)=1, but I don't think this is right to consider.


Answer



limn(1+xn)n=limn[(1+xn)nx]xnn


From
limn[(1+xn)nx]=eandlimnxnn=0,
**, we get
limn(1+xn)n=e0=1



EDIT
I add the note bellow as my calculation was considered insufficiently justified




**and because the terms are positive, and we don't have an indeterminate case 00 or 1 or 0,


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