How $U_{q}(\mathfrak{sl}_{2})$ becomes the universal enveloping algebra $U(\mathfrak{sl}_{2})$ of $\mathfrak{sl}_{2}$ 
My question is how $U_{q}(\mathfrak{sl}_{2})$ becomes the universal enveloping algebra $U(\mathfrak{sl}_{2})$ of $\mathfrak{sl}_{2}$ if we set $t=q^h$ and $q$ tends to 1.
 A: The limit is not meant to be computed with directly substituting $q=1$. It is indeterminate.  
There are various approaches for doing this computation: Formally, the general idea is to expand $t=q^h$ as a power series in $h$ and then to keep the lowest order terms (in $h$). Then you will get as the $q\to 1$ limit, your last relations (i.e. the relations of the classical enveloping algebra $U(\mathfrak{sl}_2)$). (I think that this is more or less similar to the original approach of Drinfeld). 
For more details you can see A Quantum Groups Primer, LMS, Lecture Note Series (No. 292), 2002 by Shahn Majid, ch.7, p. 42-43.
Fo another approach, based on an algebra $U^{'}_q(\mathfrak{sl}_2)$ isomorphic to $U_q(\mathfrak{sl}_2)$ for $q\neq 1$, which is however defined also for $q=1$,  you can see at C. Kassel's book on Quantum groups, ch. $VI$, sect. $2$, p. 125-126 (Prop. 2.1, 2.2). Alternatively, see this question in MO. This is (imo) more conceptual and less computational, but I'm afraid it fits less to the way you have posed your question. 
