Let `f : R to R` be a real function. The function `f` is double differentiable. If there exists `ninN` and `p in R` such that `lim_(x to oo)x^(n)f(x)=p` and there exists `lim_(x to oo)x^(n+1)f(x)` , then `lim_(x to oo)x^(n+1)f'(x)` is equal to

Answer

A

Explanation

We have `underset(xtooo)limx^(n)f(x)=p`
`implies" "underset(xtooo)lim(x^(n+1)f(x))/(x)=p`
Using L'Hospital rule, we get
`underset(xtooo)lim((n+1)x^(n)f(x)^(n+1)f'(x))/(1)=p`
`implies" "(n+1)p+underset(xtooo)limx^(n+1)f'(x)=p`
`implies" "underset(xtooo)limx^(n+1)f'(x)=-np`
Further `underset(xtooo)lim(x^(n+2)f'(x))/(x)=-np`
Using L'Hospital rule, we get
`underset(xtooo)lim((n+2)x^(n+1)f'(x)+x^(n+2)f''(x))/(1)=-np`
`implies" "-np(n+2)+underset(xtooo)limx^(n+2)f''(x)=-np`
`implies" "underset(xtooo)limx^(n+2)f''(x)=up(1+n)`