You are taking a limit so you can avoid [math] \frac{0}{0} [/math].
Remember [math] h \rightarrow 0 [/math] means that [math] h [/math] is approaching (or tends to) [math] 0 [/math] and should not be simply substituted in.
What you must remember is that you are trying to find the gradient of a tangent slope at a single point.
The general idea behind what you are doing is: You need to make a secant line between two different points, if they were the same point you could not find the slope as you would have [math] \frac{0}{0} [/math]. This however does not give us an accurate tangent slope. To overcome this problem you let the distance between those two points approach 0 (so the slope of the secant line gets ever closer to the tangent slope you are looking for), and that is where taking the limit helps us.
I suggest, as did the tree; go back and learn limits fully before trying to tackle the chain rule.