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I have been having trouble for some days now internalizing the concept of a limit especially as it appears in different contexts as in continuity,derivatives, integrals sequences etc.

Although I understand the definition of the limit in each of these instances, it just doesn't seem to be intuitively obvious, but I believe that I have zeroed in one where the problem lies.

Consider the following two statements:

We say that $\lim_{x\to c}f(x)=L$ if $f(x)$ gets closer to L as $x$ gets closer to c. $(1)$

We can make $f(x)$ arbitrarily closer to L by making $x$ sufficiently close to $c$. $(2)$

Can somenone please explain why $(1)$ and $(2)$ are equivalent?

because is seems that the $\epsilon-\delta$ definition addresses $(2)$ not $(1)$ but surely if the definition is correct it must encapsulate the intuitive although vague idea of $(1)$.

browngreen
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atifcppprogrammer
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    Statement 1 does not hold up to scrutiny. Look carefully at the graph of the function $f(x) = x\sin(1/x)$. It's an important example. $f$ approaches $0$ as $x$ approaches $0$, but $f(x)$ does not always get closer to $0$ as $x$ gets closer to $0$. This example shows that we must phrase the definition of a limit very carefully, and we can't simply use statement 1. – littleO Jul 10 '17 at 03:53
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    The two statements are not equivalent. For example, $1+1/x$ gets "closer to" $0$ as $x$ gets larger, but obviously $\lim_{x \to \infty} 1 + 1/x \ne 0,$. – dxiv Jul 10 '17 at 03:56
  • Both (1) and (2) are not rigorous enough, this is where your question arises. I recommend you back to see the true, exact and rigours definition of limits, that is $\epsilon-\delta$ formulation. – Eric Jul 10 '17 at 04:03
  • The problem here, as in many paraphrases of mathematical definition, is just what do we mean by "closer"? In the special case of where we are dealing with real numbers, what we are talking about is a metric, which is what the $\epsilon$ and $\delta$ are for. How close is "arbitrarily close"?-why, within $\epsilon$. How close does $x$ need to be to $c$ in order to ensure $f(x)$ is within $\epsilon$ of $L$? That is what $\delta$ is for. In short: rather than wave vague terms about like "close", or "close enough", we pin real number values on these terms. – David Wheeler Jul 10 '17 at 04:08
  • Statement (1) holds up to scrutiny with flying colors; see my answer below. – Mikhail Katz Jul 10 '17 at 07:21
  • In (1) it should also say "arbitrarily close"; otherwise the condition can be misinterpreted. – Mikhail Katz Jul 10 '17 at 08:04

3 Answers3

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For me, (1) is not quite true, if the adjective closer is strictly what "closer" mean. Consider $f(x)=x\sin\frac{1}{x}$, the limit of $f$ at $x=0$ is $0$, but when $x=0.01$, $f(0.01)=-0.00506366$, then when $x$ truly getting more closer to $x=0$, say $x=0.009$, then $f(0.009)=-0.00823449$, the function value leaves the $0$ more, right? So it depends on what you think the English adjective "closer" mean. If you think it mean its strictly meaning, then (1) is not a correct (equivalent) statement of limit.

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Eric
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Probably, a sensible thing to say here is this tale of Alice, Bob and Charlie.

Once upon a time, Alice read the $\varepsilon$-$\delta$ definition of limit. In order to revise her study, she had Bob help her by listening to her exposition of the concepts she had learnt. Thus phrase $(2)$ came to be. Later on, Bob tried to explain to his curious friend Charlie what they were talking about. Bob not being too subtle with either words or symbols, all he could tell Charlie was $(1)$.

By this I mean that (1) is valid just as long as you interpret (though they should technically mean something else) those "closer" as a poorly stated version of what (2) says, which on the other hand is an acceptable translation in English of the $\varepsilon$-$\delta$ definition.

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Definition (1) is completely true and in fact was the original definition of Cauchy when he spoke of continuity. It is formalized by saying that when $x$ is infinitely close but not equal to $c$, the value $f(x)$ is infinitely close to $L$. For details see Elementary Calculus.

Mikhail Katz
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  • True but it requires the formulation of an entirely new number system ! The Hyper reals ! – atifcppprogrammer Jul 10 '17 at 07:54
  • @AtifFarooq the new number system has the same elementary properties as the old one, and has a proven track record in the classroom; see this recent publication. – Mikhail Katz Jul 10 '17 at 07:55
  • It is incontestable that Abraham Robinson's treatment is far more intuitive and easier to understand but unfortunately i have to get my head round this limit treatment as it is just the way things are done where i am from – atifcppprogrammer Jul 10 '17 at 08:02
  • @Atif, you don't have to tell anybody that you understood it first via infinitesimals :-) The point is that once you develop correct intuitions via infinitesimals, you will be able to understand the epsilon-delta paraphrases better, as well. – Mikhail Katz Jul 10 '17 at 08:08
  • I will heed you advice – atifcppprogrammer Jul 10 '17 at 08:12
  • Let me know how it goes :-) – Mikhail Katz Jul 10 '17 at 08:14
  • So what about the example in the comments given by @dxiv? One has to interpret the phrase "gets closer" in a way that does not really agree with standard English in order to state that definition 1 is valid. – littleO Jul 10 '17 at 10:48
  • @little, I addressed this in my comment below the question. In both formulations you need to say "arbitrarily close". Other than that, any verbal formulation is going to be somewhat imprecise, whether it is (1) or (2). The point is that Cauchy's infinitesimal definition of continuity captures our natural intuitions better than epsilon-delta paraphrases. – Mikhail Katz Jul 10 '17 at 11:35
  • Oh I didn't notice that comment. – littleO Jul 10 '17 at 11:37