The value of k for which `f(x)={{:(,(x^(2^(32))-2^(32)x+4^(16)-1)/((x-1)^(2)),x ne 1),(,k,x=1):}` is continuous at x=1, is

Options

A. `2^(63)-2^(31)`
B. `2^(65)-2^(33)`
C. `2^(62)-2^(31)`
D. `2^(65)-2^(31)`

Answer

A

Explanation

If f(x) is continuous at x=1, then `underset(x to 1)lim f(x)=f(1)`
`Rightarrow underset(x to 1)lim f(x)=k`
`Rightarrow underset(x to 1)lim (x^(2^(32))-2^(32)x+4^(16)-1)/((x-1)^(2))=k,`
`Rightarrow underset(x to 1)lim ((x^(n)-1)-n(x-1))/((x-1)^(2))=k`
`Rightarrow underset(x to 1)lim ((x^(n)-1)/(x-1)-n)/(x-1)=k`
`Rightarrow underset(x to 1)lim ((x^(n-1)+x^(n-2)+....+x+1)-n)/(x-1)=k`
`Rightarrow underset(x to 1)lim ((x^(n)-1)+(x^(n-2)-1)+...+(x-1))/(x-1)=k`
`Rightarrow underset(x to 1)lim ((x^(n-1))/(x-1)+(x^(n-2)-1)/(x-1)+....+(x^(2)-1)/(x-1)+(x-1)/(x-1))=k`
`Rightarrow (n-1)+(n-2)+....+2+1=k`
`Rightarrow k=(n(n-1))/(2)=(2^(32)(2^(32)-1))/(2)=2^(63)-2^(31)`