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lim(n->oo)n^2(x^(1/n)-x^(1/((n+1)))),x >...

`lim_(n->oo)n^2(x^(1/n)-x^(1/((n+1)))),x >0` , is equal to (a)0 (b) `e^x` (c) `(log)_e x` (d) none of these

A

`pi`

B

`2pi`

C

`pi//2`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
`underset(ntooo)limn^(2)(x^(1//n)-x^((1)/(n+1)))=underset(ntooo)limn^(2).x^((1)/(n+1))(x^((1)/(n)-(1)/(n+1))-1)`
`=underset(ntooo)limx^((1)/(n+1))(x^((1)/(n(n+1)))-1)n^(2)`
`=underset(ntooo)limx^((1)/(n+1)).(x^((1)/(n(n+1)))-1)/((1)/(n(n+1))).(n^(2))/(n(n+1))`
`=1.log_(e)x.1=log_(e)x`
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