Answer
The precise definitions are detailed below.
Work Step by Step
(a) $\lim_{x\to 2^-}f(x)=5$
We have $\lim_{x\to 2^-}f(x)=5$ if for every number $\epsilon\gt0$, there exists a corresponding number $\delta\gt0$ such that for all $x$,
$$2-\delta\lt x\lt2\Rightarrow |f(x)-5|\lt\epsilon$$
(b) $\lim_{x\to 2^+}f(x)=5$
We have $\lim_{x\to 2^+}f(x)=5$ if for every number $\epsilon\gt0$, there exists a corresponding number $\delta\gt0$ such that for all $x$,
$$\delta\lt x\lt2+\delta\Rightarrow |f(x)-5|\lt\epsilon$$
(c) $\lim_{x\to 2}f(x)=\infty$
We have $\lim_{x\to 2}f(x)=\infty$ if for every positive real number $B$, there exists a corresponding number $\delta\gt0$ such that for all $x$,
$$0\lt |x-2|\lt\delta\Rightarrow f(x)\gt B$$
(d) $\lim_{x\to 2}f(x)=-\infty$
We have $\lim_{x\to 2}f(x)=-\infty$ if for every negative real number $-B$, there exists a corresponding number $\delta\gt0$ such that for all $x$,
$$0\lt |x-2|\lt\delta\Rightarrow f(x)\lt -B$$
