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If underset(n to oo)(lim)(e(1-1/n)^(n)-1)/(n^(alpha)), exists and is equal to l (l != 0), then the value of 12(l – alpha) is :

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Solution :Let `N = 1/X`
`l=UNDERSET(x to 0)(lim)(E(1-x)^(1//x)-1)/((1//x)^(alpha)) = underset(x to 0)(lim) (e.e^((ln(1-x))/(x))-1)/(x^(-alpha))`
`l=underset(x to 0)(lim)((ln(1-x)/(x)+1)/(x^(-alpha))) = underset(x to 0)(lim) ((-x/2-(x^2)/3......)/(x^(-alpha)))`
For limit to exist `alpha = -1`
`l = -1/2`.


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