How do we know?

Question

There are some limits when evaluated give 0/0 ,∞/∞ or some other undefined thing yet if we solve it in a particular way there does exist a limit.But some simply can't be evaluated and are always undefined as given in the solutions at the back of the book.How do we know that those "undefined" limits according to the book are really undefined and can't be solved in some particular way?Like is there some rule for it?

Answer 1

Meesam, this is a great and deep question with no quick full answer. Remember that limit as x approaches 3 means x can not be 3. Therefore lim x—> 3 (x - 3)/(x - 3) is one, is not undefined. We say a limit exists or a limit does not exist, but we do not say “ undefined “—that is for things that have no definition, such as division by zero. An example of a limit which does not exist is when the value gets larger than any positive number when we approach 3 from one side, but smaller than any negative number when we approach 3 from the other side—(1/(x - 3)) is an example. Hope this helps a little—Andrew