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Evaluate lim(xto0) ((1+x)^(1//x)-e+(1)/(...

Evaluate `lim_(xto0) ((1+x)^(1//x)-e+(1)/(2)ex)/(x^(2))`.

Text Solution

Verified by Experts

`=(1+x)^(1//x)=e^((1)/(x)log(1+x))=e^((1)/(x)(x-(x^(2))/(2)+(x^(3))/(3)-...))`
`=e^(1-(x)/(2)+(x^(2))/(3)-...)=e.e^(-(x)/(2)+(x^(2))/(3)-...)`
`=e[1+(-(x)/(2)+(x^(2))/(3)-...)+(1)/(2!)(-(x)/(2)+(x^(2))/(3)...)^(2)+...]`
`=e[1-(x)/(2)+(11)/(24)x^(2)-...]`
Hence, `underset(xto0)lim((1+x)^(1//x)-e+(1)/(2)ex)/(x^(2))=(11e)/(24)`
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