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In my high school Limit of a function was introduced as below.

We say $\lim_{x\to a^-} f(x)$ is the expected value of $f$ at $x = a$ given the values of $f$ near $x$ to the left of $a$. This value is called the left hand limit of $f$ at $a.$ We say $\lim_{x\to a^+} f(x)$ is the expected value of $f$ at $x = a$ given the values of $f$ near $x$ to the right of $a$. This value is called the right hand limit of $f$ at $a.$ If the right and left hand limits coincide, we call that common value as the limit of $f(x)$ at $x=a$ and denote it by $\lim_{x\to a} f(x)$.

For initial examples we first compute the value of $f$ at points very close to $a$ and then deduce the limit. Then move to polynomials. Then if functions under consideration are rational functions. We first evaluate these functions at the prescribed points. If this is of the form $\frac{0}{0}$ , we try to rewrite the function cancelling the factors which are causing the limit to be of the form $\frac{0}{0}$ .

Now in undergraduate studies we were introduced to the $\epsilon- \delta$ definition of limit. I understand it fully. I just needed an example of a quick example showcasing how the high school definition of limit is incomplete and is not formal enough. An example that illustrates why we need formalisation of the phenomenon that as $x$ approaches $a$, $f(x)$ approaches its limit. Can anyone please provide such an example?

user534666
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    The first definition is just not precise. What does "expected" mean? – Karl Jun 17 '23 at 07:32
  • What do you mean by "expected vaue of the function"? – Zima Jun 17 '23 at 08:26
  • @ Hello I am referring to page 284 here https://ncert.nic.in/ncerts/l/kemh113.pdf So expected values means the value that f(x) approaches as x approaches a. – user534666 Jun 17 '23 at 09:00
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    @user534666. How do you define "the value that f(x) approaches as x approaches a"? – md2perpe Jun 17 '23 at 09:06
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    The NCERT textbook isn't trying to define limit, but trying to give some idea about the limit concept without using a precise definition. Most intro calculus textbooks do that. The problem is that they don't mention that it isn't a definition and it isn't precise. – Paramanand Singh Jun 17 '23 at 09:41
  • @ParamanandSingh I wanted some motivation with an example to transition from this idea/ intuition to the formal $\epsilon delta$ definition. Showing why we need formalisation. – user534666 Jun 17 '23 at 09:54
  • Your definition has got the left and right the wrong way round. It should be: "$\lim_{x\to a^+} f(x)$ is the expected value of $f$ at $x = a$ given the values of $f$ near $x$ to the right of $a$. This value is called the right hand limit of $f$ at $a.$" But other than that and the poor choice of the word "expected", the essence of the question is actually interesting. I will try to think of a (counter-)example. – Adam Rubinson Jun 17 '23 at 09:56
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    @user534666. Please prove that $\lim_{x \to 0} x \sin x = 0.$ – md2perpe Jun 17 '23 at 10:01
  • @AdamRubinson Thank you. I have made the correction. Actually the above book linked in the comment is the Basic Math Textbook in India and this is how the chapter on limits was taught. https://ncert.nic.in/ncerts/l/kemh113.pdf page 284 – user534666 Jun 17 '23 at 10:20
  • Well, we do need a precise definition of limit if we need to prove theorems related to limits. On the other hand if we accept theorems on limits without proof then we can use them to evaluate limits without any appeal to intuition. IMHO that is what a typical calculus beginner should do. Compare the situation to using Heron's formula for area of a triangle without the proof for the formula. – Paramanand Singh Jun 17 '23 at 10:56
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    This is a possible duplicate, or at least closely related question: https://math.stackexchange.com/questions/947677/the-need-for-the-formal-definition-of-a-limit – Adam Rubinson Jun 17 '23 at 10:56

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This may be the kind of example you want. Suppose $f(x)=x$ if $x$ is rational, $f(x)=17$ if $x$ is irrational, and you want to determine whether $\lim_{x\to0}f(x)$ exists. So, you look at what happens at values of $x$ near zero. Well, if you make life easy on yourself and only look at rational values of $x$ near zero, you'll conclude the limit exists, and equals zero. If your best friend is more imaginative, and only looks at irrational values of $x$ near zero, she'll conclude that the limit exists, and equals $17$. And you're both wrong, since the limit doesn't exist at all.

For a more complicated function, it may be harder to see that there's one collection of $x$-values where the function does one thing, and a different collection where the function does something very different. That's why you need epsilons and deltas, so you can look at all values of $x$, in a systematic way.

Gerry Myerson
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