Actually, there's an even more formal and general (and therefore hard to follow) definition that takes it beyond functions of a single real variable, which generalizes the idea of a delta to "open balls" or "neighborhoods." If you didn't find " Formal Definition of a Limit" in our archives, you may want to read it, because it attempts to demystify the formal definition.
The latter is just a precise statement of the former. The general idea of "approaching," and the formal definition using delta and epsilon, are the two main ways to discuss limits. Are there any other ways of describing a limit?
It just happens, as we’ll see, that the primary purpose of limits in calculus, the derivative, is a case where you can’t (just) plug in a number that’s the interesting case. Should the concept of a limit be applicable to all functions, or is it really only useful for certain situations?Īs I suggested in my mention of continuity, the limit concept is applicable to all functions, but is only "interesting" for those peculiar functions that either are not continuous, or are not defined at some points.įor a continuous function, evaluating a limit requires just evaluating the function (once you know it really is continuous). I’m not sure all calculus students need to be able to prove limits (that’s for the math majors), but seeing and understanding such a proof is intended to help clarify the point of limits, which is, in part, that approaching a value is not the same as actually having that value. As Josh recognized, some limit questions, like \(\displaystyle \lim_\), are important and non-obvious but others seem hardly worth doing, because you can just plug in the number. This happens in many fields of math: to keep it simple, we have to give problems where the answer seems almost trivial. In case you haven't been introduced to continuity yet, a continuous function is one whose limit is in fact equal to its value the fact that most familiar functions are continuous is the reason the concept seems a little silly to you. It is possible that such a function might NOT approach its value at 3, so you have to prove that it does. But also, it introduces the concept of continuity. I think it's important to have not only a good understanding of the formal definitions, but also a feel for how things work and what they are all about.Įxactly: This is for practice with the concept of limits and the methods for proving them. These are good questions, and too easily overlooked when this subject is taught too formally. So, beyond the mechanics of finding or proving limits, what is the concept really about? Josh basically understands what a limit is, but sees it as unnecessary or trivial in many cases. A service like this is very useful and informative. I really appreciate any time you spend on this. So why the limit? Why not just say, "when h = 0"? The definition of a derivative is:īut I don't see the need for a limit here because all you do is plug in the value of 0 for h once you've reduced the function into its simplest form. Should the concept of a limit be applicable to all functions, or is it really only useful for certain situations?Ģ. Is this just for practice, or am I missing something?ġ. I can see the need for a limit on things like:īecause you need to know what the value "approaches" not what it actually equals, because at 0 it basically doesn't exist. "The value that f(x) approaches as x approaches some number." Most of the time I get the same vague description: I searched all over your site and couldn't find anything about understanding limits.īasically I'm trying to find any extra help I can on understanding the concept of a limit. The following question arrived in 2001: Understanding the Need for Limits What good are limits? Why did they have to be invented? Are they as simple as they seem? Why is an epsilon-delta proof necessary? Are limits useless? But too often the concept is not sufficiently motivated. Many calculus courses start out with a chapter on limits or they may be introduced in a “precalculus” course.
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