If `("lim")_(xvec0)({(a-n)n x-tanx}sinn x)/(x^2)-0,` where `n` is nonzero real number, the `a` is 0 (b) `(n+1)/n` (c) `n` (d) `n+1/n`

Options

A. 0
B. `(n)/(n+1)`
C. n
D. `n+(1)/(n)`

Answer

D

Explanation

If f(x) is continuous at x=0, then `underset(x to 0)lim f(x)=f(0)`
`Rightarrow underset(x to 0)lim (((a-n)nx-tanx)sin nx)/x^(2)=0`
`Rightarrow underset(x to 0)lim n{(a-n)n-(tanx)/(x)}(sin n x)/(nx)=0`
`Rightarrow n((a-n)n-1)xx1=0`
`Rightarrow (a-n)n-1=0 Rightarrow a=n+(1)/(n)`