Let `f(x)=(3)/(4)x+1,f^(n)(x)` be defined as `f^(2)(x)=f(f(x)),` and for `n ge 2, f^(n+1)(x)=f(f^(n)(x))." If " lambda =underset (n to oo)(lim)f^(n)(x),` then show that λ is independant of x.

Options

A. `lambda` is independent of x
B. `lambda` is a linear polynomial in x
C. the line `y=lambda` has slope 0
D. the line `4y=lambda` touches the unit circle with center at the origin.

Answer

A, C, D

Explanation

`f^(2)(x)=f((3)/(4)x+1)=(3)/(4)((3)/(4)x+1)+1=((3)/(4))^(2)x+(3)/(4)+1(1)`
` f^(3)(x)=f{f^(2)(x)}=(3)/(4){f^(2)(x)+1}`
`=(3)/(4){((3)/(4))^(2)x+(3)/(4)+1}+1`
`=((3)/(4))^(3)x+((3)/(4))^(2)+(3)/(4)+1`
` :. f^(n)(x)=((3)/(4))^(n)x+((3)/(4))^(n-1)+((3)/(4))^(n-2)+ ... +((3)/(4))+1`
`=((3)/(4))^(n)x+(1-((3)/(4))^(n))/(1-(3)/(4))`
` :. lambda=underset(n to oo)(lim) f^(n)(x)=0+4=4`