`Let f(x)` be a polynomial satisfying `lim_(xtooo) (x^(2)f(x))/(2x^(5)+3)=6" and "f(1)=3,f(3)=7" and "f(5)=11.` Then
`lim_(xto1) (x-1)/(sin(f(x)-2x-1))` is equal to

Answer

A

Explanation

Since `underset(xtooo)lim(x^(2)f(x))/(2x^(5)+3)=6` which is finite and non-zero, `f(x)` must be polynomial of degree '3'.
Also `f(1)=3,f(3)=7" and "f(5)=11`
`:." "f(x)=lambda(x-1)(x-3)(x-5)+(2x+1)`
`implies" "underset(xtooo)lim(x^(2)(lamda(x-1)(x-3)(x-5)+2x+1))/(2x^(5)+3)=6`
`implies" "(lamda)/(2)=6`
`implies" "lamda=12`
`:.f(x)=12(x-1)(x-3)(x-5)+(2x+1)`
So, `f(0)=12(-1)(-3)(-5)+1=-179`
`underset(xto1)lim (x-1)/(sin(f(x)-2x-1))`
`=underset(xto1)lim(x-1)/(sin[12(x-1)(x-3)(x-5)])`
`=underset(xto1)lim(12(x-1)(x-3)(x-5))/(sin[12(x-1)(x-3)(x-5)]).(1)/(12(x-3)(x-5))`
`=(1)/(96)`