Question

Let f: n⁺ to n⁺ be a function satisfying the relation f( x. f (y))= f (x .y) + X for all xy belonging to R then

Lim. [ ( f(x))^1/3-1]÷[(f(x))^1/2-1]=______

X----->0

Answer 1

Given relation is f(x.f(y))=f(xy)+x (1.56)
Interchanging x and y in Eq. (1.56), we have
f(y.f(x))=f(yx)+y (1.57)
Again replacing x with f(x) in Eq. (1.56) we get
f(f(x).f(y))=f(y.f(x))+f(x) (1.58)
Therefore, Eqs. (1.56) - (1.58) imply
f(f(x).f(y))=f(xy)+y+f(x) (1.59)
Again interchanging x and y in Eq. (1.59), we have
f(f(y).f(x))=f(yx)+x+f(y) (1.60)
Equations (1.59) and (1.60) imply
f(xy)+y+f(x)=f(yx)+x+f(y) (1.61)
Suppose f(x)−x=f(y)−y=λ
Substituting f(x)=λ+x in Eq. (1.56), we have
x.f(y)+λ=(xy+λ)+x
⇒x.f(y)=xy+x
Therefore x(y+λ)=xy+x[∵f(y)=λ+y]
⇒λ=x
⇒λ=1(∵x>0)
So f(x)=x+λ=x+1
Hence limx→0(f(x))13−1(f(x))12−1=limx→0(1+x)1/3−1(1+x)1/2−1=limx→0((1+x)1/3−11+x−1).(1+x−1(1+x)1/2−1)=1/31/2=23