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Let f(x)=(lim)(n rarr oo)sum(r=0)^(n-1)x...

Let `f(x)=(lim)_(n rarr oo)sum_(r=0)^(n-1)x/((r x+1){(r+1)x+1})` . Then (A) `f(x)` is continuous but not differentiable at `x=0` (B) `f(x)` is both continuous but not differentiable at `x=0` (C) `f(x)` is neither continuous not differentiable at `x=0` (D) `f(x)` is a periodic function.

A

f(x) is continuous but not differentiable at x = 0

B

f(x) is both continuous and differentiable at x = 0

C

f(x) is neither continuous not differentiable at x = 0

D

f(x) is a periodic function

Text Solution

Verified by Experts

The correct Answer is:
C
`t_(r+1)=(x)/((rx+1){(r+1)x+1})`
`=((r+1)x+1-(rx+1))/((rx+1)[(r+1)x+1])`
`=(1)/((rx+1))-(1)/((r+1)x+1)`
`therefore" "S_(n)=sum_(r=0)^(n-1)t_(r+1)=1-(1)/(nx+1)`
`therefore" "f(x)=1-(1)/(nx+1),x ne0`
`" "f(0)=0 and`
`therefore" "underset(nrarroo)(lim)S_(n)=underset(nrarroo)(lim)(1-(1)/(nx+1))=1`
`therefore" "f(0^(+))=1`
Thus, f(x) is neither continuous nor differentiable at x = 0.
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