Tuesday, 22 November 2016

convergence divergence - Limit of the function limxto0(fracex1x)



I'm trying to solve this limit without the use of L'Hospital, but I'm doing something wrong. The limit should be:



limx0(ex1x)=1



My attempted proof:
limx0(ex1x)=limn(e(1n)11n)=limn(n(e(1n)1)1)=limn(n(e01)1)=limn(01)=0



I assume the mistake is that I've used the continuity of the exp function.


Answer



A variation of Bernard's answer:



ex1x=1exxex=1exex1xx011(ex)x=0=e0=1


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