# What exactly are limits?

So as far as I know, a limit of a function is the $y$ value that we get as $x$ gets closer and closer to the limit but remains distinct from the limit.
Also, limit when approached from right should be the same as limit when approached from left or else the limit does not exist.

The problem I have is with limits involving complex functions. I was watching a video on this and the presenter said that if we were to make a circle of radius $1$ around the $x$ value we input into the function, the $y$ value will be in the same neighborhood.

Can someone please tell me what that is?

• I think your question can be best addressed by recommending you to take a basic calculus course, as there hardly exists an answer out of the blue without resourcing to the basics. – DonAntonio Jun 22 '13 at 14:24

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.

• Ok, so if I am getting it right --- the circle shows that the limit exists or not. Basically, if the limits from both the sides agree, they will fall within the circle. Else, they will not :-) – Little Child Jun 22 '13 at 17:08
• that's right. but be careful. that circle thing is really just a simple idea to make a point. sometimes your left and right limits will be in the same area graphically, but won't converge on the same number. the limit exists if and only if the left hand limit = right hand limit. – Tyler Murphy Jun 25 '13 at 14:18