Question

Why the definition of limit is necessary to Calculus?

Answer 1

are you asking why we need limits?

Answer 2

I'm asking why some one needs to say I give you an epslon near this L (epson<f(x,y)-L) if you give me a delta (>0) near to (x,y).

Answer 3

epsilon is not near L...

we want f to be near L (n some open ball around L...a ball of radius epsilon)

Answer 4

Do you agree that concepts need a definition in order to talk about them in a precise way? I mean, if there is no "formal" definition for limit and it comes down to intuition... what's to say your intuition is the same as mine?

Answer 5

If you agree that we need some kind of definition, how would you define a limit?

Answer 6

I was talking to a friend and we agree that this definition bring all points (acumulation points) near (even if the point is not defined in the domain, we can get as close as we want to x without actually reaching it.) to be "part" of the domain. So we can bring these points that are not in the domain, but near the domain to our world of calculus. Without this definition we could not use them. Am I right?

Answer 7

So, in calculus we are trying to get x/y where both of them get infinitely small at different rates, or sometimes the same rate, but the point is, the rates are related because y is a function of x, that part is important. So imagine we have two balls, and they keep getting smaller and smaller, so small we can't measure how small they are, infinitely small. But, the one thing we do know is the ratio of the size of the big ball to the small ball. As we make both of the balls smaller and smaller that ratio gets more and more stable. The concept of a limit is a definition, it allows the delta epsilon proof. With this proof, we can show that it is always the case that for any epsilon (how close a function is to the limit), I can produce a delta that will give us that epsilon. So in calculus, there are really infinitesimal errors when we say that x^2's derivative is 2x, but if you ask me "What about those errors" those errors are really how close the function is to that limit (epsilon), and every time you say well, we have an error of X, I can keep producing deltas that make the error less than X. I can make the error infinitely small, it approaches a limit of zero. So any time someone tries to see that error, it shrinks out of sight. No matter how exact of an answer you want, I can make the error smaller than that exactitude demands.

Answer 8

kbomeisl, what is small or infinitely small? It depends on how I want this small. In terms of galaxy small could be the distance between the moon and earth. So this definition bring to real world these abstract concepts?

Answer 9

So, infinitely small is not a relative concept, it is an absolute quantity with absolute properties like the number 2. It is simply the opposite of infinitely large. The cardinality (number of elements in) the set of all the natural numbers is infinity, specifically the smallest class of infinity named aleph null. One divided by aleph null would be the first class of infinitesimals. There isn't really a name for it.
Here's how we define infinity, every time you give me a number, I will tell you infinity is larger. Every concept is math is just a natural logical consequence of a number of axioms. The natural numbers. Axiom 1: 1 is in the set Axiom 2: if n is in the set, then n+1 is in the set. Boom, that's it, I don't need an essay to tell you what the natural numbers are, they are just anything that fulfills those two axioms. Similarly, what is infinity. x, such that x>n, where n is all real numbers. Infinitely small things are the opposite, 1/infinity, if you like. They simply fulfill the axiom: y<n, n=all real numbers. In the real world, it hovers right in between zero and the smallest real number.
Math is a realm of pure ideas, the limit pops out of the concept of the real numbers in pure logic. If you look up the axioms behind the real numbers, the Peano axioms, you will notice, after some heavy analysis, that they generate the real numbers we know and love, 3,5,pi, e, 1/6, etc, but they also generate a bunch of new concepts we have never seen before that still have all the properties of the real numbers, but are smaller than any of them. These numbers are qualitatively different in many respects, we call them the infinitesimals. They are just as real as the "real numbers", and were rigorously proven as cogent object in their own right hundreds of years after calculus. You are quite right to be suspicious of the concept of a limit! The existence of infinitesimals as proven above makes the concept rigorous.

Answer 10

More directly responding to your question, it is indeed smaller than the distance between the moon and the earth.

Answer 11

kbomeisl, thank you for your time and atention, the question about infinitesimal was in my mind for a long time. Some people tried to explain me what is infinitesimal but I think they were lost in the question like me. Again thank you!

Answer 12

My pleasure, it’s a smart question to ask. Calculus is just algebra without limits, most people gloss over what that concept actually means. It seems rather unnatural, but in fact the limit really came out of the very essence of nature. Newton saw F=ma got more and more exact the smaller the interval (change in velocity)/(time) got, and that nature really worked at the infinitely small level (but you know, don’t tell quantum mechanics that). The laws of thermodynamics, gravity, electromagentism, even quantum, don’t care about finite time or position changes, they actually only act on infinitesimal quantities. We only perceive finite sized time intervals or lengths and masses, etc, but physics is actually acting only on infinitesimal time lengths, or masses, that we cannot perceive (that’s where the limit lives). That’s why all the laws of physics are partial differential equations. Now here is why your question is exactly on the right track to thinking the right way about deep mathematical mysteries:
The concept of these infinitesimals was engineered by Newton and Leibniz so that they would be so small they would in fact allow us to dodge any “real” “finite” error. So they were invented and defined to be so small that they were incapable of producing any error logically or practically. But they actually have earthshattering consequences! We weren’t trying to prove the existence of infinitesimal numbers when we did this, it was an accident. We were trying to prove that the real numbers were a first order logical system. The importance of this is that many important mathematical results only apply to first order logic, like the compactness theorem, completeness (Godel), etc. Results that are not only practically beautiful and useful, but if they fail, madness ensues. Because there were always infinitesimals dancing around, the proof of the compactness theorem for the real numbers fails (I can always use the infinitesimals to undermine certain parts of the proof), and so does the completeness theorem. But more importantly, the the real numbers we trust are secure and rigorously proven are now a second order system. Godel Incompleteness applies, which is the Heisenberg Uncertainty for math, but way way worse. In a second order system, there are always things that are true, but cannot be proven (always), and there are things that are false but cannot be disproven (always), and they are false because they can be proven. See John Barwise, A Handbook on Mathematical Logic, for the details here. But the gist is that you are asking the right questions. For a more down to earth example of how infinitesimals are very real and have a tangible effect on calculations, see the Dirac delta potential. It is a little bump that is infinitesimally wide but infinitely high, the result is a finite number. Because of that little bump a quantum wavefunction will actually be radically different, different energy levels, different positions, momenta, etc.