MHB When Can We Determine if a Limit Exists or Not in Calculus?

[find_the_fun]

What does it mean for a limit to exist or not exist? I'm reviewing improper integrals and I forget what it means.

 

[Ackbach]

There are a number of ways a limit could fail to exist:

1. For the usual two-sided limit, if the limit from the left and the limit from the right do not agree, the two-sided limit just plain d.n.e. (does not exist).

2. Any infinite limit, whether positive or negative infinity, whether two-sided or one-sided, can also be said not to exist.

3. Sometimes a function starts oscillating wildly near a point and doesn't settle down to anyone value. $\sin(1/x)$ at the origin is one such example.

No doubt there are other ways, but these are some of the more common ways a limit can fail to exist.

 

[find_the_fun]

There are a number of ways a limit could fail to exist:

1. For the usual two-sided limit, if the limit from the left and the limit from the right do not agree, the two-sided limit just plain d.n.e. (does not exist).

2. Any infinite limit, whether positive or negative infinity, whether two-sided or one-sided, can also be said not to exist.

3. Sometimes a function starts oscillating wildly near a point and doesn't settle down to anyone value. $\sin(1/x)$ at the origin is one such example.

No doubt there are other ways, but these are some of the more common ways a limit can fail to exist.

If there are other methods of determining whether a limit exists or not that means it's not actually the definition. What DOES it mean for a limit to exist (not just how to tell)? Thanks.

 

[Ackbach]

If there are other methods of determining whether a limit exists or not that means it's not actually the definition. What DOES it mean for a limit to exist (not just how to tell)? Thanks.

Right; you can use the limit laws in quite a few cases to tell whether a limit exists or not. Here is an excellent link you can read that will help you understand how to find whether limits exist, and what the value is if it does (those two processes are often the same thing!). You can also check out my http://mathhelpboards.com/calculus-10/differential-calculus-tutorial-1393.html.