Changing limit depending on question being asked?

Rough depiction of the relevant graph for this question.

My question pertains to finding the limit of [f(x) / g(x)] as x approaches 2. I know that this cannot be done because g(x) equals 0 as x approaches 2, and we can't divide f(x) by that. As such, we are to look at the limit of both f(x) and g(x) as x approaches 2 from the left and from the right. This is where I start to get confused.

f(x) is no problem: as x approaches 2 from the left and right, the limit of f(x) appears to be 1. For g(x), however, I'm not certain. I would assume that the limit of g(x) as x approaches 2 from BOTH the right and left would be 0 given that g(x) is continuous within domain (1, infinity). However, in class, we were told that the limit of g(x) as x approaches 2 from the left/negative is actually negative infinity, and that the limit of g(x) as x approaches 2 from the right/positive is infinity.

Why is this? If we were asked the limit of g(x) as x approaches 2 from the left or right in the context of most any other question, the answer would be 0. Why is it not 0 here?

Hopefully that makes sense, any input would be appreciated.