The Uniqueness of The Limit

A very important theorem you need to learn is exactly this.

By the very way it works, if a function has a limit at a point, it can not have any other limits at that single point. Any limit that exists at any point of a function is unique. If it wouldn't be unique, it wouldn't be a function at all.

If the limit at a point of a function exists, no other limit can exist at that point, and this limit is unique.

That's it, really.
Let's go on to learn about Limits towards infinities