If the (for example) left-hand limit of a function approaches infinity because there is a vertical asymptote. Does the limit exist?

No, if the left-hand limit fo a function approaches infinity because there is a vertical aymptote the limit does not exist.

Explanation:

Important: infinity is not a value!

When it is said that a limit approaches infinity, it means that the function becomes indefinitely large: as you get closer and closer to the input value, the function becomes larger and larger; in this case, no value can be assigned to the limit, that is, the limit does not exist

For there to be a limit, the function has to approach a specific value. Infinity is not a number; it does not indicate a particular value; again, the limit does not exist.