4
$\begingroup$

This question already has an answer here:

The intuitive definition for $\lim\limits_{x \to a} f(x) = L$ is the value of $f ( x )$ can be made arbitrarily close to $L$ by making $x$ sufficiently close, but not equal to, $a$.

the formal definition of limit:For every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < | x − a | < δ implies | f(x) − L | < ε.

Why we need the formal definition of a limit? Does the intuitive definition has some flaw ?

I understand the intuitive definition quite well, but I don't understand much about the formal one.

P.S.I must declare I only have some basic knowledge of limit ,I started to learn calculus a few days ago .

Understanding of the formal and intuitive definition of a limit, This post is not good ,because it covered more than one question , I picked one question from it ,so we can discuss it more targeted .

$\endgroup$

marked as duplicate by Will Orrick, Rahul, Mark Fantini, Claude Leibovici, Hans Lundmark Sep 27 '14 at 9:03

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • 4
    $\begingroup$ If you are precise about what you mean by arbitrary and sufficient, I think you will see that your intuition is the same as the formal definition. $\endgroup$ – illysial Sep 27 '14 at 1:25
  • $\begingroup$ I understand the intuitive definition quite well ,but I don't understand much about the formal one . $\endgroup$ – iMath Sep 27 '14 at 1:31
  • 6
    $\begingroup$ Someone could rightly ask you: "what do you mean by arbitrarily close to $L$"? You might say: "I mean for any "distance" $\epsilon >0$ around $L$." Well isn't this the same as saying that $|f(x)-L|< \epsilon$? Similarly by sufficiently close to $a$ you mean for some distance $\delta>0$ around $a$. Well we write "distance around a less than $\delta$" as $|x-a|< \delta$ $\endgroup$ – illysial Sep 27 '14 at 1:37
  • $\begingroup$ Define the words "arbitrary" and "sufficient" in precise mathematical terms. If we are to move on and do greater things with our bag of mathematical facts and tools, we need total precision. Without that we are just building a house of cards. At this point in your studies, it is more important that you understand the intuitive definition of limit, but attempting to understand the formal definition is well worth your while, and over time, if you keep trying to understand it, you may well succeed. $\endgroup$ – J. W. Perry Sep 27 '14 at 5:55
  • $\begingroup$ does “making x sufficiently close, but not equal to, a .” correspond to “there exists a real δ > 0 such that for all real x, 0 < | x − a | < δ” ? this is the point where I cannot understand ,because I am not sure if “there exists a real δ > 0 such that for all real x, 0 < | x − a | < δ” shows "x close enough, but not equal,to a". specifically ,I don’t think it shows the dynamic process of “$x$ get closer and closer to a ” ? $\endgroup$ – iMath Sep 27 '14 at 8:51
6
$\begingroup$

The formal definition is very similar to your intuition, which isn't surprising. The main difference is, consider the function $$f(x)=\sin\left(\frac{1}x\right).$$ as $x\rightarrow 0$. It exhibits increasingly rapid oscillations which don't vanish in amplitude as $x$ decreases towards $0$. Now, we could, for any $L$ in $[-1,1]$ choose an arbitrarily small $x$ such that $f(x)=L$. Your definition would seem to allow that we could thus say $\lim_{x\rightarrow 0}f(x)=L$ for any value in $[-1,1]$, which is obviously problematic. A more accurate english definition would be:

$\lim_{x\rightarrow a}f(x)=L$ if we can force the value of $f(x)$ to be arbitrarily close to $L$ by restraining the value of $x$ to lie sufficiently close to $a$.

which does not befall the same problems - it is, in fact, the formal definition. The idea here is that we exclude oscillatory behavior - we are saying that we can, around $a$, draw a little box about the point $(a,\lim_{x\rightarrow a}f(x))$ which contains the function.

$\endgroup$
  • $\begingroup$ To be precise we exclude unbounded oscillatory behavior, e.g. $x^2\sin(1/x)$ oscillates but still has a limit as $x\to 0$. But anyway, I like this answer. $\endgroup$ – David Z Sep 27 '14 at 7:02
  • $\begingroup$ does “there exists a real δ > 0 such that for all real x, 0 < | x − a | < δ” in the formal definition show “making x sufficiently close, but not equal to, a .” ? or in your words ,does it show "restraining the value of x to lie sufficiently close to a" ?this is the point where I cannot understand ,because I am not sure if “there exists a real δ > 0 such that for all real x, 0 < | x − a | < δ” shows "x close enough, but not equal,to a". specifically ,I don’t think it shows the dynamic process of “x get closer and closer to a ” ? $\endgroup$ – iMath Sep 27 '14 at 9:15
  • $\begingroup$ It's not immediately true from the definition that x needs to get closer and closer to a - like, for a constant function f(x)=c proving that some limit is c works, since no matter what δ we choose, f(x) is in (c-ε,c+ε) over (a-δ,a+δ). It's probably more productive to imagine the strip of values L-ε<y<L+ε, which narrows, and check if there is always some leeway around a such that any x there has f(x) in the band. $\endgroup$ – Milo Brandt Sep 27 '14 at 14:38
6
$\begingroup$

To understand why we need the formal definition of a limit, you really have to understand why we need careful definitions at all in mathematics. A formal definition is really needed when intuition is insufficiently precise to avoid not only inconsistencies, but more importantly, the depth of understanding that allows us to see counterexamples to the intuitive definition that we would otherwise be blind to.

A very good example for the beginning calculus student is the relationship between continuity and the existence of the derivative-2 critical concepts in calculus you may or may not be yet aware of. Intuitively, a continuous function is one where there are no gaps in the graph of the function-that is,the function is defined on the entire domain. For example, the function f(x) = ax+b, where a,b are real numbers, is continuous. f has values defined on the entire real line-it defines a straight line in the plane. Where a= 2 and b=5,the graph is: Graph of 2x+5

Now let's consider the function f(x) = 1/x. This function is not continous at x=0 and it's graph looks like this: Graph of 1/x

Clearly,there's a "hole" in the graph at x=0. But as you can also see,the graph of the function otherwise has no holes or gaps. If we exclude x = 0 from the domain of the function, (1/x) is continuous on it's domain by our "intuitive" definition. This is important,keep it in mind.

The related concept of the derivative is a measure of the rate of change of a function at a specific point, geometrically actualized by a tangent line to the curve at a specific point x= a at f(a).For example, given the function f(x) = x^2, the derivative is 2x and geometrically this gives:

Tangent to parabola

Notice the derivative is a local property of f-i.e. it's defined at a point only. At most points on a continuous curve in the plane,it should be possible to construct a tangent line. So intitively, you would think that a function which was continuous would be differentiable i.e. the derivative would exist-at most points. Indeed, most mathematicians before the 19th century didn't worry about the existence of a derivative as long as the function's graph was sufficiently "smooth". So imagine the bewilderment in the 19th century when B.Bolzano and Karl Weierstrass both produced examples of functions which were everywhere continous on thier domains-and nowhere differentiable! To make the point, here's a computer generated graph of the Weierstrass function: enter image description here

This is a function most mathematicians wouldn't have thought was possible given the "intuitive" definitions and machinery they'd been working with since Newton. If you think these are merely mathematical abstractions that have no relevance in the "real" world, you'd be very wrong indeed-it turns out many nonlinear phenomena, such as turbulence, exhibit precisely these kinds of mathematical behaviors locally! So it really is critical to have precision in mathematics to prevent our intution from leadong us astray. Intuition combined with rigor, in my opinion, is the most powerful human method of generating both mathematical and scientific facts.

That answer your question?

$\endgroup$
0
$\begingroup$

Of course, the $\delta, \epsilon$-definition is unintuitive because we have that for all... exists... such that... for all... This definition was designed so that to can work at low level, so is not usually used in calculus (maybe, only in the introduction), but is usefull in analysis, to prove many properties about real numbers a their relation with derivative and integration.

You can check the about the construction of reals number by Cauchy sequences, to get a better idea about the definition.

For example, the definition of a real number

A real number is defined to be an object of the form $LIM_{n \to \infty} a_n$, where $(a_n)_{n=0}^\infty$ is a Cauchy sequence of rational numbers.

A sequence $(a_n)_{n=0}^\infty$ of rational numbers, $a_0, a_1, a_2, \dotso$, is a Cauchy sequence if and only if for every $\epsilon > 0$, there exists an $N \ge 0$ such that $|a_j - a_k| \ge \epsilon$ for all $j, k \ge N$.

So in the decimal system $$1.4, 1.41, 1.414, 1.4142, 1.41421, \dotso$$ and $$1.5, 1.42, 1.415, 1.4143, 1.41422, \dotso$$ are sequences of rationals that eventualy close (converge) to the same real number $\sqrt{2}$.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.