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I understand the meaning of epsilon transitions, but could someone give example where epsilon transition becomes handy?

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Epsilon transitions come in handy to `chain' languages.

For example: to construct the kleene closure of a language, one connects the accepting states to a new starting state with epsilon transitions and one connects this new starting state with the old starting state with an epsilon transition.

This construction is probably a lot harder when one is not allowed to use epsilon transitions.

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    In the same vein, $\epsilon$-transitions provide a simple proof that the union of, say, regular languages is regular.2012-09-02