L = {x / |x| = 3}. Is this language regular?

L = { x / |x| = 3 }

Assume that x belongs to alphabet {0,1}. I think the above language is regular. A DFA can be used to determine the above language. Am i correct? Is the above language regular?

If this language L is regular, then it should satisfy pumping lemma. Then there exist w = xyz, where y can be raised to any power of n >= 0. And still the resulting string would be in the language L.

But on the other hand, if we pump more letters then the resulting string will not be in the language. The language L only accepts string of length 3.

Pumping Lemma states that for every regular language there exists an integer p, such that string w of at-least length p can be written as w = xyz and y can be pumped.

Here are my doubts.

  1. Is this language L regular?
  2. If so does it satisfy Pumping Lemma?
  3. Pumping Lemma states that every regular language has a pumping length p >=1. Does this language does not have one?