Constant functions:

Here we want f(x) = 7 for example. We want to say $\lim_{x\rightarrow 5} f(x) = 7$. So for every t > 0 we have to produce m_t > 0 so that if 0 < |x-5| < m_t then |f(x) - 7 | < t. This is not hard:

|f(x) - 7| = |7 - 7| = 0 < t

is true regardless of x, so m_t could be any positive real number. We can take m_t =1.