how to demonstrate that $\lim_{x\to2}3x^2=12$ using $\gamma$ and $\epsilon$ definition how to demonstrate that $\lim_{x\to2}3x^2=12$ using $\gamma$ and $\epsilon$ definition? 
My current steps I have in order from top to bottom:
$|3x^2-12|<\epsilon$ and $0<|x-2|<\gamma$
assume $z=x-2$
$0<|z|<\gamma$ and $3z^2+12|z|<\epsilon$
$3z^2+12|z|<3\gamma^2+12\gamma=3\gamma(4+\gamma)$
Then assume $\gamma<1$
$\gamma=15$ hence $\gamma=\frac{\epsilon}{15}$
then now i am stuck and have no clue how to proceed

What you have given is sometimes called a "synthetic" proof- you start with what you want to prove, $|3x^2- 12|< \epsilon$, and arrive at $\gamma\le \frac{\epsilon}{15}$.  Of course, you can't prove something by assuming it is true!  This is, still, a valid proof because every step is reversible.  That is, the "real proof" starts by taking $\gamma$ to be any number less than $\frac{\epsilon}{15}$ then works backwards to $|3x^2- 12|< \epsilon$.
