The point the guy in your lecture was making was right in line with what you were already understanding about limits. Take for example the limit(X->0) of 1/|x|.
If you approach X from the left, the y value approaches infinity. Thus, the left-hand limit is infinity.
If you approach x from the right, the y value is also infinity. Thus, the right-hand limit is infinity.
Since Left-hand limit = right-hand limit, the two limits are equal.
Now consider what the professor said about a circle. If you draw a circle around 1/|X| where X is really close to zero (can't use zero for obvious reasons) and X is positive, ten your circle will enclose both the left-hand side of the graph and the right. That's what he means by being in the same neighborhood.
For a counter example, consider f(x) = 1/x. As we approach from the left, the limit is negative infinity. As we approach from the right, the limit is positive infinity. If you now draw your circle of radius 1 somewhere near X=0 for x positive, it will not incorporate the left-hand side of the function because it is negative. Thus, the left-hand and right-hand limits are not in the same neighborhood. Therefore, the limit does not exist.