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## Homework Statement

Suppose a function satisfies the conditions

1. f(x+y) = (f(x)+f(y))/(1+f(x)f(y)) for all real x & y

2. f '(0)=1.

3. -1<f(x)<1 for all real x

Show that the function is increasing throughout its domain. Then find the value:

Limit

_{x -> Infinity}f(x)

^{x}

## The Attempt at a Solution

I proceed by putting x,y=0 in eq 1.

I get the following roots for f(0)={-1,0,1}

But if I take f(0)={-1,1}, f(x) will become a constant function and will be equal to +1 when f(0)=1 and -1 when f(0)=-1, thereby violating condition 3

So f(0)=0

From equation 1: I assume 'y' as a constant and differentiate wrt x

f ' (x+y)=(f ' (x)(1-f

^{2}(y))) / (1+f(x)f(y))

^{2}

I put x=0;

I get f ' (y)=1-f

^{2}(y) Using condition 3; I prove that the derivative is always positive.

I have been able able to solve the first part of the question. But I couldn't evaluate the limit

Limit

_{x -> Infinity}f(x)

^{x}. Please help me on the limit part.

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