Thursday, 23 May 2013

How to Solve This Exponential Limit without Derivate / L'Hôspital's Rule



can someone teach me how can I solve this limit without using the L'Hopital's Rule?



limx0(2+x22x2)1x2




Thanks in advance.


Answer



2+x22x2=1+2x22x2=1+x21x22


and y=x21x220, as x0. Hence
(1+x21x22)1x22x2=(1+y)1/ye,


as x0.



Finally
(2+x22x2)1/x2=((1+x21x22)1x22x2)11x22e,


since, if f(x)e and g(x)1, then f(x)g(x)e.


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