As for the mathematician, I used to be strong in maths, but got bored of them, more interested in the theory than the practice part of it honestly XD

For the answer to the question though, I'm working on it... been a long time since I've had to do this kind of problem. The square root exponent isn't so much the problem, as the annoying variables and lack of anything consistant in the problem is.

Still, lemme see...

((4 + h)^0.5 - 2)/h

(4+h)^0.5 -2 = h

4+h -2^2 = h^2

h=h^2 <-- at this point it basically is either h=1 or h=0 being the only two possibilities...

h^0.5 = h

(1h)^0.5 = 1h

1^0.5 = 1 <-- is this step even allowed? O.o

1 = 1^2

1 = 1

I have nooooooooo clue how accurate that is and I'm relatively sure I screwed something up somewhere, but I'm really not sure. It's been aaaaaaages since I've had to play with that kind of maths sadly.

But it came out to a whole number in the end so I'm going to hope that's right XD

EDIT!

DURR! I just realized after posting that I screwed something up that should've saved me a few steps XD

h=h^2

h/h = (h^2)/h

1 = 1^2

1 = 1

Same answer but that makes more sense somehow.

(To give an idea of how long it's been, I had to doublecheck on the definitions of solve, rationalize and simplify...)