What is the purpose of the pumping lemma for regular languages with epsilon transitions in automata theory?

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What is the purpose of the pumping lemma for regular languages with epsilon transitions in automata theory?

The purpose of the pumping lemma for regular languages with epsilon transitions in automata theory is to provide a tool for proving that a language is not regular. It states that for any regular language L, there exists a pumping length p such that any string s in L with length greater than or equal to p can be divided into five parts, s = xyzuv, satisfying certain conditions. By showing that these conditions cannot be satisfied for a given language, we can conclude that the language is not regular.