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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

`-log2`

C

1

D

e

Text Solution

Verified by Experts

The correct Answer is:
B
`L=underset(xrarr0)(lim)sum_(n=1)^(oo)((-1)^(n)f((x)/(n)))/(x)`
`=underset(xrarr0)(lim)sum_(n=1)^(oo)((-1)^(n)(1)/(n)f'((x)/(n)))/(1)" (Applying L'Hospital Rule)"`
`=sum_(n=1)^(oo)((-1)^(n)(1)/(n)f'(0))/(1)`
`=-sum_(n=1)^(oo)((-1)^(n-1))/(n)`
`=-log2`
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