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If underset(n to oo)(lim)(e(1-1/n)^(n)-1...

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 :

A

4

B

3

C

6

D

7

Text Solution

Verified by Experts

The correct Answer is:
C
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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