Let `f(x)=lim_(nrarroo) sum_(r=0)^(n-1)(x)/((rx+1){(r+1)x+1})`. Then

Options

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

Answer

C

Explanation

`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.