Chain rule proof

Hello guys I have read the chain rule proof on the website and have understood it but at the bottom it shows a 'Flawed Proof' and I do not understand why that proof is Flawed.

http://math.rice.edu/~cjd/chainrule.pdf

It's something to do with limits or dividing by zero??

I expect it is because $\displaystyle \displaystyle g'(x)$ could be $\displaystyle \displaystyle 0$, which means you wouldn't be able to multiply by $\displaystyle \displaystyle \frac{1}{g'(x)}$.

Also, I believe that the $\displaystyle \displaystyle h$ that has been used to calculate $\displaystyle \displaystyle f'(g(x))$ is different to the $\displaystyle \displaystyle h$ that is used to give $\displaystyle \displaystyle g'(x)$. Since they are not the same, you shouldn't cancel them... Instead, they should have been given two different names, e.g. $\displaystyle \displaystyle h_1, h_2$...