My text defines left limit inferior and superior as follows:
If $ f$ is bounded on $ [a,x_0)$: Let $ S_f(x; x_0) = \sup\limits_{x \leq t < x_0} f(t) $, $I_f(x; x_0) = \inf\limits_{x \leq t < x_0} f(t) $.
Then the left limit superior = $ \lim\limits_{x \to x_0^-} S_f(x; x_0) $, left limit inferior = $ \lim\limits_{x \to x_0^- } I_f(x; x_0) $.
However there is no mention of a right limit superior or right limit inferior, and I was wondering how one could define them in a similar manner.