My question is part of a bigger one that I'm using in a graphical proof. I have two functions, tanh(ap) and 2a/(1+a^2). The a is the input and p is a constant. I'm trying to find the value for p which is the bridging point between the two functions having 1 intersection and 2 intersections. To do this, I said that we want to find the value for p which makes tanh(ap) greater than 2a(1+a^2) for every point on the interval (0,1) (because the second function's max is at 1, so the two intersections will be on the sides of a=1)
So I have tanhap>2a/(1+a^2)>0, but I'm not sure how to go about solving it. I can't set them equal to each other at a=1 because tanha never reaches 1 so I get an undefined answer. Is there something I can do with limits?