I need to build a DFA from $\varepsilon$NFA, so first of all I assume that I need to convert the $\varepsilon$NFA to NFA and then to DFA (because I know the method of converting from NFA to DFA, and we learn how to remove $\varepsilon$'s from $\varepsilon$NFA and make it NFA.But the I'm still stuck at some cases....For example:See this $\varepsilon$NFA:I want to make it NFA (I don't know if to write the whole algorithm...), and I'm not sure which is right:This: Or this (at the comments.... I can't put more then 2 images)I think that the first is the right one, but I'm not sure...At the second one - the number of the 0-s must be even to be at $q_1$ but at the original $\varepsilon$NFA it can be any number of 0-s...One more thing, it's correct that I'm not putting 0,1 insted of the $\varepsilon$? (Because 1 from $q_0$ brings me to $\varnothing$...) 
How to make a NFA (or DFA) from $\varepsilon$NFA?
automata;finite automata
I think the following should work. For each state q, define E(q) to be the set of all states reachable from q by following zero or more ε-transitions. Then, create a new NFA as follows:The set of states in the NFA is the set $\{ E(q) | q \in Q \}$.The start state is $E(q_0)$.For each state $E(q)$ in the new NFA, the transitions are given as follows: for each state $r \in E(q)$ that has a transition to a state $s$ on a character $a$, add a transition from $E(q)$ to $E(s)$ on $a$.The accepting states in the NFA are the states containing at least one accepting state in the original NFA.The basic idea behind the construction is to have each NFA state correspond to a set of states in the ε-NFA. This effectively simulates taking all possible ε-transitions at each point in time.