I have never seen this way of writing before, we have always written $\lim_{x\rightarrow 0^{-}}$ for the left side and $\lim_{x\rightarrow 0^{+}}$ for the right side of limit. Or we have worked with $<,>$.
So what does it mean and does it even exist...? I couldn't even find it in latex which explains why I have used a picture. Very ridiculous if people want make your life harder by always finding new ways of notating something. As if there weren't enough already.
$x\nearrow 2$ typically means "$x\to 2$ in an increasing fashion" (and therefore, in particular it implies $x\to 2^-$).
Similarly, $x\searrow 2$ typically means "$x\to 2$ in an decreasing fashion" (and therefore, in particular it implies $x\to 2^+$).
As mentioned in the comments below, the $\nearrow 2$, $\searrow 2$ notations (and their less $LaTeX$-savvy equivalents $\uparrow$/$\downarrow$) make more sense for sequences, where the monotonicity implication they carry is a stronger statement than just writing $\to 0^+$ or $\to 0^-$. For the "real-analysis" (non-sequential) version, they are basically equivalent, so feel free to use the one you (or your peers) like most.