Let us define $U_q(sl(2))$ as the algebra with four generator as usual $$K\,K^{-1}=K^{-1}K=1,$$$$K\,E\,K^{-1}=q^{2}E,\,K\,F\,K^{-1}=q^{-2}F,\\EF-FE=\frac{K-K^{-1}}{q-q^{-1}}.$$ I want to easily show that can be thought as a deformation of $U(sl(2))$. So the easiest thing to do is to think $K=q^H$ and derive for example $$K\,E\,K^{-1}=q^{2}E,$$ which becomes $$q^H\,E\,q^{-H}=q^{2}E,$$ deriving by $q$ we obtain $$Hq^{H-1}\,E\,q^{-H}-q^{H}\,E\,Hq^{-H-1}=2qE,$$ and letting $q \rightarrow 1$ (but let us suppose that $q$ is never 1 just to avoid the problems in the definition of the algebra). As limit you get the usual commutation relations $$HE-EH=2E.$$
Now my questions: Since I never saw this derivations in any book I assume something is wrong with this procedure, I would like to understand what is wrong with all this step taken? What is missing to make this argument rigorous? As said before let us suppose that $q$ is never equal to 1 to avoid that type of problem that I'm already aware of...
Thank you in advance