Does g(x + h) = g(x) + (h g'(x) ) as h approach 0 (i get it from (g(x+h)-g(x))/h = g'(x) ).if i am wrong,...

Does g(x + h) = g(x) + (h g'(x) ) as h approach 0 (i get it from (g(x+h)-g(x))/h = g'(x) ).if i am wrong,
how do i rewrite g(x+h)=?

They aren't equal, but they are approximately equal.

Since [g(x + h) - g(x)]/h ≈ g'(x) (as h --> 0, this becomes exact), we can multiply both sides by h to get:
g(x + h) - g(x) ≈ h*g'(x) ==> g(x + h) ≈ g(x) + h*g'(x).

You cannot multiply both sides of:
lim (h-->0) [g(x + h) - g(x)]/h = g'(x),

to get what you had.

I hope this helps!