`("lim")_(xvecoo)n^2(x^(1/n)-x^(1/((n+1)))),x >0,i se q u a lto` 0 (b) `e^x` (c) `(log)_e x` (d) none of these

Options

A. `pi`
B. `2pi`
C. `pi//2`
D. none of these

Answer

C

Explanation

`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`