What kind of "mathematical object" are limits?

Question

When learning mathematics I tend to try to reduce all the concepts I come across to some matter of interaction between sets and functions (or if necessary the more general Relation) on them. Possibly with some extra axioms thrown in here and there if needed, but the fundamental idea is that of adding additional structure on sets and relations between them. I've recently tried applying this view to calculus and have been running into some confusions. Most importantly I'm not sure how to interpret Limits. I've considered viewing them as a function that takes 3 arguments, a function, the function's domain and some value (the "approaches value") then outputs a single value. However this "limit function" view requires defining the limit function over something other then the Reals or Complexes due to the notion of certain inputs and outputs being "infinity". This makes me uncomfortable and question whether my current approach to mathematics is really as elegant as I'd thought. Is this a reasonable approach to answering the question of what limits actually "are" in a general mathematical sense? How do mathematicians tend to categorize limits with the rest of mathematics?

Answer 1

I'm not sure what flaws the other answer is finding with your idea. This is a perfectly interesting question. For better or worse, the real answer is a mathematical object is whatever type you can make of it. One thing can be a different type in different contexts, so it is quite useful to ask, in general, "Can I write this another way?"

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You do want to be careful with wording though: a "limit" is just a value, whatever value you get. But you're asking about taking a limit. There's no reason you can't conceive of that as a function. (Typically a map that takes functions as input and gives numbers as output are called functionals, and Functional Analysis is a major field of math.) There is also no problem with mapping things to infinity. (That's evidenced by limits doing just that! Check out the extended reals on Wikipedia.)

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Of course, the implicit question you may be asking is whether this conception is useful. Not off the top of my head, but people have found uses for the functional F_p(f) := f(p), which takes a function and gives its value. So maybe you're on to something! The general philosophy is "if you want to understand something, hit it with a function". So perhaps the question is not whether something is a function, but what other functions can tell you. Moreover, if thinking of limits in this way helps you because that's the framework you have already built for yourself, then it is useful. Elegance is overrated in the math community.

Answer 2

I generally tell my students that, in laymen terms, to say lim as x-->c of f(x) = L just means that when x is close to c, BUT NOT EQUAL TO C, then f(x) is close to L. This is easy to view graphically, and matches up with any function that is continuous at x = c, or has a removable discontinuity at x = c.

Thus, the limit is just a y-value (for functions of one variable) that your function approaches, though the function may or may not take on the value of L when x = c.

To fully understand limits of functions of one variable, review the epsilon-delta definition carefully.

Though the definition of limits expands in a natural way to multi-variable functions, evaluating limits for functions of 2 or more variables can get much more complicated. Understanding the relationships between the geometric and algebraic properties of limits also plays a huge role, as with many mathematical topics.

It is good to delve into the topic of limits. It is the foundation of all calculus. The definitions of the derivative, and of the anti-derivative, of a function are both limits, and being able to apply calculus to physics requires an understanding of the underlying geometry of limits.

Answer 3

Well your inquiry indicates some flaws in the conceptualization of limit.

In the "real' world two difference concepts of limit exist.

The first is something that should not be exceeded, e.g., a speed limit.

The second is something that actually stops and action, e.g., $10 is my limit.

The mathematical concept of limit is related to infinite geometric series.

0.3333333333333....... = ?

∞

∑ (0.3)(0.1)^n-1

n=1

Now we all know that the "answer" is 1/3, Yet it is 1/3 only when we do not stop the series to determine the sum of the series!

Therefore, in this case we say that 1/3 is the limit.

For any δ you name I can name an n such that

|1/3 - ∑(0.3)^n-1| < δ

This is the definition that should have been presented in your pre-calculus class.

Answer 4

The limit is the same as any other value in the set. If you want to see a full treatment of limits, you should study real analysis. I think the more abstract concept of a limit is a "limit point" which is a point that the values nearby approach.

However, one way out of your confusion is that "infinity" is not really a limit. If the limit approaches infinity, we actually say the limit doesn't exist. I know sometimes we distinguish limits as "DNE", infinity, or negative infinity, but those are just different classifications of a limit that doesn't exist. So you may have a hard time thinking of infinity as a limit because limits are only values that are part of the set (in this case, real numbers).