I am currently re-reading and re-learning with basic concepts in calculus/analysis. I have tried to prove certain limits exist using the epsilon delta definition. Many of these examples are fairly easy because they are made so to guide new students like me to choose the right $\delta$. The gist of these problems is that illustred through this example:

Prove $\displaystyle\lim_{x\to5}|3x+2|=17$.

Let $\epsilon>0, \epsilon$ is arbitrary, pick $\delta=\epsilon/3\implies3\delta=\epsilon$. Then $\forall x\in\mathbb{R}$, then within $0<\color{red}{|x-5|}<\delta$, we have $|3x+2-17|<|3x-15|=3\color{red}{|x-5|}$. This makes $|f(x)-L|$ equal to $|x-a|$, and the $3$, let call it a constant C, they make it easy to choose the value of $\epsilon$ because now you can come to the conclusion that $3|x-5|<3\delta\implies3|x-5|<\epsilon/3\implies|x-5|<\epsilon$.

Stewart does provide a harder problem where such a $C$ is not provided, like $\displaystyle\lim_{x\to3}x^2=9$. I will ask the question on this problem in another thread, but here I wish to ask you something.

I have been spoonfed like many students in Calc 1 with a list of limit law to calculate limits. This means, for example, when x approaches $3$, $x^{2}$ approaches $9$. The term approach seems to be forgotten since for simple limit of polynomial, we can just plug the limit in and it pops out the value.

My point is with the definition of epsilon delta, the gist seems to lie in the concept of neighbourhood or nearness. I have seen the title of a book by Dugac and it contains the word "voisinage" in French. This makes me think that we are interested in the neighbour of the limit point, rather than the exact limit point.

I have also read that the definition of limit is independent from the concept of function being defined. A classic example is that $$\displaystyle\lim_{x\to0}\dfrac{\sin(x)}{x}=1$$, despite the fact that $x$ is not defined at $x=0$. But the limit still exists. There are many proofs to this, most useful is trigonometric proof.

I have looked into cases when limit doesn't exist, and this site suggests that:


The one-sided limits are not equal

The function doesn't approach a finite value (see Basic Definition of Limit).

The function doesn't approach a particular value (oscillation).

The x - value is approaching the endpoint of a closed interval

Of all these, the ones with one-sided limit makes me wonder. Since in one dimension, we can easily define left and right, in 3 dimensions, this may not be very useful, am I correct?

So the point of epsilon and delta is to generalize the concept of neighbourhood or nearness of the limit point, because this can be generalized to higher dimensions?

Sorry, I haven't studied multivariate calculus so my experience lies entirely within function of one variable.

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    $\begingroup$ "I have also read that the definition of limit is independent from the concept of function being defined" Exactly; this is the reason why it is not always so simple as "just plug the limit in and it pops out the value": in some case there is no value of $f$ in $x_0$ but still there is a limit. $\endgroup$ – Mauro ALLEGRANZA Feb 9 at 15:42
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    $\begingroup$ IMO, the epsilon-delta mechanism is not motivated by "higher dimensions"; it is simply the way to give a precise mathematical definition of "approaching to". In simple case of one variable with continuous functions, this is exactly what happens when we move along the graph of the function approaching $x_0$: at a certain point we arrive at $x_0$ and we find $f(x_0)$. $\endgroup$ – Mauro ALLEGRANZA Feb 9 at 15:43
  • $\begingroup$ There is a nice nuance to the concept of limit. What is the integral value of each limit? Are they just idealized number of a bunch of non ending decimals? $\endgroup$ – James Warthington Feb 9 at 15:44
  • $\begingroup$ It depends if the limit is rational or irrational. Rational numbers are ending decimals or periodic decimals; irrational numbers are non-ending decimals :) What does it matter? The limit is some real number, that's all we care for now. $\endgroup$ – peter.petrov Feb 9 at 15:46
  • $\begingroup$ If we deal with real-valued functions, the limit is a real number (a point on the number line). $\endgroup$ – Mauro ALLEGRANZA Feb 9 at 15:46

"the point of epsilon and delta is to generalize the concept of neighborhood or nearness of the limit point, because this can be generalized to higher dimensions"

I would say it's not to generalize it but to define it. Usually the delta pertains to $x$, the epsilon pertains to $f(x)$.

In higher dimensions one defines something called an open ball. Well, that is a neighborhood in higher dimensions.

  • $\begingroup$ @peterpetrov I asked a friend of mine who is an accomplished Master student and I said that the epsilon delta definition is not very intuitive, he counters that it is highly intuitive. I guess that's because he can view things from a higher vantage point. $\endgroup$ – James Warthington Feb 9 at 15:46
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    $\begingroup$ No, it's just a matter of getting used to it. If you keep playing with it in 5-10 years it will look intuitive to you too, I am pretty sure :) Many years ago it looked non-intuitive to me too. But it's a powerful technique, allows us to formalize all these proofs. $\endgroup$ – peter.petrov Feb 9 at 15:47
  • $\begingroup$ why do we choose this conception and not infinitesimal which is more intuitive? Is it because the space we are dealing with may not be $\mathbb{R}$ but some other spaces? Hence the power of a more abstract definition? $\endgroup$ – James Warthington Feb 9 at 15:50
  • $\begingroup$ Exactly, the limit would be the center of this increasingly small open ball in $\mathbb{R}^n$ which has an infinity of values of $f$ as $x \in \mathbb{R}^n$ approaches the interest value. That's my geometric intuition whenever I work with a limit of a function. $\endgroup$ – Leandro Abib Feb 9 at 15:50
  • $\begingroup$ In that sense, you force the value to be as close as you want to the limit. $\endgroup$ – Leandro Abib Feb 9 at 15:51

Limits are tricky in this sense. It is true that the limit $$ \lim_{x \to 3}x^2 $$ can be evaluated by simply plugging in $x=3$, but the reason for this is not as simple as you might think. I will explain this in more detail in a moment, but first let's consider what limits are in general. As you have already correctly noted, limits tell us about the behaviour of a function as you approach a value, but not when you actually 'get there'. The example you gave is perfect. When we write $$ \lim_{x \to 0}\frac{\sin x}{x} = 1 $$ what we mean is that as $x$ approaches $0$, $\frac{\sin x}{x}$ approaches $1$. In more detail, $\frac{\sin x}{x}$ gets arbitrarily close to $1$. Pick a tiny number, say $0.000001$. Provided that $x$ is sufficiently close to $0$, the difference between $\frac{\sin x}{x}$ and $1$ will be smaller than that tiny number. This helps us state the precise definition of limit:

We say that the limit as $x$ approaches $0$ of $\frac{\sin x}{x}$ equals $1$ because we can make $\frac{\sin x}{x}$ arbitrarily close to $1$ by requiring that $x$ is sufficiently close to, but unequal to, $0$.

Hopefully, this is the gist of the definition of a limit that you were looking for. And the precise definition definition with epsilon's and delta's is saying essentially the same thing. It just clarifies our wordy description even further. Note that the last section of our wordy definition was

...requiring that $x$ is sufficiently close to, but unequal to, $0$.

As we mentioned earlier, limits tell us about what happens we get close to, but never actually reach a certain value. In a way, the limit tells us what we anticipate when we get close to a value, rather what actually happens there. The function $\sin x / x$ is undefined when $x=0$. It makes no sense to actually ask what happens when we 'get to zero'. But we can make meaningful statements about what happens when we get close to zero. Let's go back to your example, $$ \lim_{x \to 3}x^2 \, . $$ If limits are all about what happen when we get close to $3$, then why can we simply plug in $x=3$? This is because the function $x^2$ (and all polynomial functions in general) are continuous. A function is said to be continuous at the point $x=a$ if $$ \lim_{x \to a}f(x) = f(a) \, . $$ In other words, the anticipated value is the same as the value actually obtained by plugging in $x=a$. Many functions do not have this property. In the earlier example, $$ \frac{\sin x}{x} $$ is not even defined at $x=0$, and so it doesn't even make sense to ask whether it is continuous there. Nevertheless, it can be shown that all polynomial functions are continuous everywhere, and so evaluating the limit $$ \lim_{x \to a}P(x) $$ where $P$ is an arbitrary polynomial, simply amounts to plugging in $x=a$. This is what is so confusing to newcomers to calculus. Although limits tell you about the behaviour of a function as you get close to a point, a lot of the time you can take a shortcut and just plug in the value being approached. This is because, with a little work, we can show that the 'elementary functions' (polynomials, the exponential function, sine and cosine, etc.) are continuous at the point $x=a$ if they are defined there. But don't let that mislead you. When we say that $\lim_{x \to a}f(x) = L$, we are still talking about the behaviour of a function as you get close to, but unequal to $a$; however, in some instances it is possible to work out this limit simply by plugging, provided that you know beforehand that the function is continuous.

  • $\begingroup$ let me thank you for being so passionate and patient in responding to my question, I am reading carefully right now. $\endgroup$ – James Warthington Feb 9 at 16:21
  • $\begingroup$ Now I think limit is a formalized concept of various numerical computational experiences. $\endgroup$ – James Warthington Feb 9 at 16:23
  • $\begingroup$ @JamesWarthington What do you mean by 'various numerical computational experiences'? $\endgroup$ – Joe Feb 9 at 16:28
  • $\begingroup$ I mean before the time we have the concept of limit of a function, mathematicians may need numerous computations to know what is the value of the function at a given point. That's just my guess. $\endgroup$ – James Warthington Feb 9 at 16:30
  • $\begingroup$ @JamesWarthington That's not really the case. Limits are useful in a number of ways: finding derivatives, better understanding the properties of functions, etc. but I don't see a use for them in numerical computation. Anyway, if you have any questions regarding my answer, then please let me know. I'll be happy to help. $\endgroup$ – Joe Feb 9 at 16:32

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