This answer is wrong as shown by the OP, but I thought his comment and the link he gave are valuable, so I leave this answer intact:
This isn't really a big theorem, but in $\mathbb R$, if the left hand and the right hand limits exist and are equal, then the limit exists. This idea dissipates once you go to higher dimensions (even just $\mathbb R^2$). There's no more left and right limits, you need to approach the limit point through any path. With that being said, maybe there's a theorem that states that if all those limits exist, then the limit exists. I don't know how useful that theorem would be though.
I tried thinking of more substantial examples before submitting, but couldn't find any. I haven't necessarily seen every single-variable theorem generalized to higher dimensions, but the few I thought of seem to be easily generalized. I'm intrigued now to see what are some other, bigger results that can't be generalized.