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find_the_fun
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What does it mean for a limit to exist or not exist? I'm reviewing improper integrals and I forget what it means.
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 said: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 said: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.
A limit is a mathematical concept that describes the behavior of a function as its input approaches a certain value or point. It is used to determine the value that a function approaches as the input gets closer and closer to a specific point.
A limit does not exist when the function does not approach a single value as the input approaches a specific point. This can happen for several reasons, such as a jump or a hole in the graph of the function, or oscillation around the specific point.
To determine if a limit does not exist, you can use the limit laws to evaluate the left and right limits at the specific point. If the left and right limits are not equal, or if they approach different values, then the limit does not exist.
An example of a limit that does not exist is the function f(x) = 1/x, as x approaches 0. The left limit is -∞ and the right limit is +∞, so the limit does not exist.
Understanding when a limit does not exist is important because it helps us to identify and analyze discontinuities in a function. It also allows us to determine the behavior of a function at specific points, which can be useful in many mathematical and scientific applications.