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

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Discussion Overview

The discussion revolves around the concept of limits in calculus, specifically focusing on the conditions under which a limit exists or does not exist. Participants explore definitions, examples, and methods related to limits, including improper integrals and oscillating functions.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks clarification on what it means for a limit to exist, particularly in the context of improper integrals.
  • Another participant outlines several scenarios where a limit may fail to exist, including discrepancies between left-hand and right-hand limits, infinite limits, and functions that oscillate without settling to a single value, such as $\sin(1/x)$ at the origin.
  • A repeated point emphasizes that if there are multiple methods to determine the existence of a limit, it raises questions about the definition of a limit itself.
  • One participant suggests that limit laws can be used to ascertain whether a limit exists and offers a link for further reading on the topic.

Areas of Agreement / Disagreement

Participants express differing views on the definition of a limit and the implications of having multiple methods to determine its existence. The discussion remains unresolved regarding a clear definition of what it means for a limit to exist.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about definitions of limits and the reliance on various methods for determining limit existence, which are not fully explored.

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.
 
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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.
 
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.
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.
 
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.

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.
 

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