Question

Is the limit of a function as you approach a point always the same if you approach from either side? Would that hold true for vertical asymptotes, or would the limit be infinity and negative infinity?

Answer 1

left and right limits do not always approach the same point, and yes vertical asymptotes can cause one to go to infinity and the other to approach negative infinity.

Answer 2

simplest example,
\[f(x)=\frac{1}{x}\] limit as x goes to zero from the left is minus infinity, from the right infinity