(click on that to see the code)
Since exponentials are always positive, we always have the numerator smaller than the denominator, in absolute value. That tells us that the fraction is always between -1 and 1.
As x goes to infinity, goes to 0 so tanh(x) goes to 1. As x goes to negative infinity, goes to 0 so tanh(x) goes to -1.
I know lim tanh (x)=+-1 as x tends to infinity. How do i prove tanh(R) = (-1,1)