Thursday, 27 August 2015

calculus - What is? limxtoinftyleft(fraceleft(1+frac1xright)xright)x





Find limx(e((1+1x)x))x.







limx(e((1+1x)x))x=limx(ee)x=limx1x=



Why is this incorrect?


Answer



Take the logarithm,




log(e(1+1x)x)x=xloge(1+1x)x=x(1xlog(1+1x))=x(1x(1x12x2+O(x3)))=x(11+12x+O(x2))=12+O(x1).



And hence




limx(e(1+1x)x)x=e1/2.


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