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Let f : R rarr R be a differentiable fun...

Let `f : R rarr R` be a differentiable function at x = 0 satisfying f(0) = 0 and f'(0) = 1, then the value of `underset(x rarr 0)(lim) (1)/(x) . underset(n = 1)overset(oo)(sum)(-1)^(n).f((x)/(n))`, is

A

`0`

B

`-In 2`

C

1

D

e

Text Solution

Verified by Experts

The correct Answer is:
B
`underset(x to 0)lim (1)/(x) underset(n=1)overset(oo)sum (-1)^(n)f((x)/(n))`
`=underset(x to 0)lim underset(n=1)overset(oo)sum (-1)^(n)f((x)/(n)) (1)/(x)`
`underset(n=1)overset(oo)sumxx ((-1)^(n))/(n) underset(x to 0)lim (f((x)/(n))-f(0))/((x)/(n))`
`underset(n=1)overset(oo)sum ((-1)^(n))/(n) f'(0) underset(n=1)overset(oo)sum ((-1)^(n))/(n)="In 2"`
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