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

1.	Evaluate `lim_(xto0) ((1+x)^(1//x)-e+(1)/(2)es)/(x^(2))`.
Answer» `=(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)`

Discussion

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