Finding any $\epsilon$ vs finding minimal $\epsilon$ for case 3 of Master theorem
First, note that $\log_4 3 = 0.792...$.  Now the key observation is that $n^{0.792..}$ grows asymptotically slower than $n \lg n$, i.e., $n^{0.792...} = O(n \lg n)$, or if you prefer, $n \lg n = \Omega(n^{0.792...})$.  Why?  Well, one way to see it is to notice that $n^{0.792...} = O(n)$ and $n = O(n \lg n)$: $n^{0.792...}$ grows asymptotically slower than $n$, because the exponent is smaller; and $n$ grows asymptotically slower than $n \lg n$.

What's up with the $\epsilon=0.2$? That's probably a clunky way of getting at the fact that $n^{0.792...}$ is smaller than $n^{1.000...}$.  In particular, $0.792 + 0.2 &lt; 1.0000$.  Personally, I don't find it a particularly helpful explanation for why $n^{0.792...} = O(n)$, so I would suggest you ignore it.  If you find it confusing, it's not just you, and it doesn't represent some shortcoming in your understanding; it just seems like a confusing way to explain it to me.