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(ε, δ)-definition of limit

From Wikipedia, the free encyclopedia

Whenever a point x is within δ units of c, f(x) is within ε units of L
Whenever a point x is within δ units of c, f(x) is within ε units of L

In calculus, the (ε, δ)-definition of limit ("epsilondelta definition of limit") is a formalization of the notion of limit. The concept is due to Augustin-Louis Cauchy, who never gave an () definition of limit in his Cours d'Analyse, but occasionally used arguments in proofs. It was first given as a formal definition by Bernard Bolzano in 1817, and the definitive modern statement was ultimately provided by Karl Weierstrass.[1][2] It makes rigorous the following informal notion: the dependent expression f(x) approaches the value L as the variable x approaches the value c if f(x) can be made as close as desired to L by taking x sufficiently close to c.

YouTube Encyclopedic

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  • ✪ Epsilon-delta limit definition 1 | Limits | Differential Calculus | Khan Academy
  • ✪ Calculus - Find the limit of a function using epsilon and delta
  • ✪ Epsilon-delta definition of limits
  • ✪ Finding delta from a graph and the epsilon-delta definition of the limit (KristaKingMath)
  • ✪ Definition of a Limit Epsilon Delta Proof 3 Examples

Transcription

Let me draw a function that would be interesting to take a limit of. And I'll just draw it visually for now, and we'll do some specific examples a little later. So that's my y-axis, and that's my x-axis. And let;s say the function looks something like-- I'll make it a fairly straightforward function --let's say it's a line, for the most part. Let's say it looks just like, accept it has a hole at some point. x is equal to a, so it's undefined there. Let me black that point out so you can see that it's not defined there. And that point there is x is equal to a. This is the x-axis, this is the y is equal f of x-axis. Let's just say that's the y-axis. And let's say that this is f of x, or this is y is equal to f of x. Now we've done a bunch of videos on limits. I think you have an intuition on this. If I were to say what is the limit as x approaches a, and let's say that this point right here is l. We know from our previous videos that-- well first of all I could write it down --the limit as x approaches a of f of x. What this means intuitively is as we approach a from either side, as we approach it from that side, what does f of x approach? So when x is here, f of x is here. When x is here, f of x is there. And we see that it's approaching this l right there. And when we approach a from that side-- and we've done limits where you approach from only the left or right side, but to actually have a limit it has to approach the same thing from the positive direction and the negative direction --but as you go from there, if you pick this x, then this is f of x. f of x is right there. If x gets here then it goes here, and as we get closer and closer to a, f of x approaches this point l, or this value l. So we say that the limit of f of x ax x approaches a is equal to l. I think we have that intuition. But this was not very, it's actually not rigorous at all in terms of being specific in terms of what we mean is a limit. All I said so far is as we get closer, what does f of x get closer to? So in this video I'll attempt to explain to you a definition of a limit that has a little bit more, or actually a lot more, mathematical rigor than just saying you know, as x gets closer to this value, what does f of x get closer to? And the way I think about it's: kind of like a little game. The definition is, this statement right here means that I can always give you a range about this point-- and when I talk about range I'm not talking about it in the whole domain range aspect, I'm just talking about a range like you know, I can give you a distance from a as long as I'm no further than that, I can guarantee you that f of x is go it not going to be any further than a given distance from l --and the way I think about it is, it could be viewed as a little game. Let's say you say, OK Sal, I don't believe you. I want to see you know, whether f of x can get within 0.5 of l. So let's say you give me 0.5 and you say Sal, by this definition you should always be able to give me a range around a that will get f of x within 0.5 of l, right? So the values of f of x are always going to be right in this range, right there. And as long as I'm in that range around a, as long as I'm the range around you give me, f of x will always be at least that close to our limit point. Let me draw it a little bit bigger, just because I think I'm just overriding the same diagram over and over again. So let's say that this is f of x, this is the hole point. There doesn't have to be a hole there; the limit could equal actually a value of the function, but the limit is more interesting when the function isn't defined there but the limit is. So this point right here-- that is, let me draw the axes again. So that's x-axis, y-axis x, y, this is the limit point l, this is the point a. So the definition of the limit, and I'll go back to this in second because now that it's bigger I want explain it again. It says this means-- and this is the epsilon delta definition of limits, and we'll touch on epsilon and delta in a second, is I can guarantee you that f of x, you give me any distance from l you want. And actually let's call that epsilon. And let's just hit on the definition right from the get go. So you say I want to be no more than epsilon away from l. And epsilon can just be any number greater, any real number, greater than 0. So that would be, this distance right here is epsilon. This distance there is epsilon. And for any epsilon you give me, any real number-- so this is, this would be l plus epsilon right here, this would be l minus epsilon right here --the epsilon delta definition of this says that no matter what epsilon one you give me, I can always specify a distance around a. And I'll call that delta. I can always specify a distance around a. So let's say this is delta less than a, and this is delta more than a. This is the letter delta. Where as long as you pick an x that's within a plus delta and a minus delta, as long as the x is within here, I can guarantee you that the f of x, the corresponding f of x is going to be within your range. And if you think about it this makes sense right? It's essentially saying, I can get you as close as you want to this limit point just by-- and when I say as close as you want, you define what you want by giving me an epsilon; on it's a little bit of a game --and I can get you as close as you want to that limit point by giving you a range around the point that x is approaching. And as long as you pick an x value that's within this range around a, long as you pick an x value around there, I can guarantee you that f of x will be within the range you specify. Just make this a little bit more concrete, let's say you say, I want f of x to be within 0.5-- let's just you know, make everything concrete numbers. Let's say this is the number 2 and let's say this is number 1. So we're saying that the limit as x approaches 1 of f of x-- I haven't defined f of x, but it looks like a line with the hole right there, is equal to 2. This means that you can give me any number. Let's say you want to try it out for a couple of examples. Let's say you say I want f of x to be within point-- let me do a different color --I want f of x to be within 0.5 of 2. I want f of x to be between 2.5 and 1.5. Then I could say, OK, as long as you pick an x within-- I don't know, it could be arbitrarily close but as long as you pick an x that's --let's say it works for this function that's between, I don't know, 0.9 and 1.1. So in this case the delta from our limit point is only 0.1. As long as you pick an x that's within 0.1 of this point, or 1, I can guarantee you that your f of x is going to lie in that range. So hopefully you get a little bit of a sense of that. Let me define that with the actual epsilon delta, and this is what you'll actually see in your mat textbook, and then we'll do a couple of examples. And just to be clear, that was just a specific example. You gave me one epsilon and I gave you a delta that worked. But by definition if this is true, or if someone writes this, they're saying it doesn't just work for one specific instance, it works for any number you give me. You can say I want to be within one millionth of, you know, or ten to the negative hundredth power of 2, you know, super close to 2, and I can always give you a range around this point where as long as you pick an x in that range, f of x will always be within this range that you specify, within that were you know, one trillionth of a unit away from the limit point. And of course, the one thing I can't guarantee is what happens when x is equal to a. I'm just saying as long as you pick an x that's within my range but not on a, it'll work. Your f of x will show up to be within the range you specify. And just to make the math clear-- because I've been speaking only in words so far --and this is what we see the textbook: it says look, you give me any epsilon greater than 0. Anyway, this is a definition, right? If someone writes this they mean that you can give them any epsilon greater than 0, and then they'll give you a delta-- remember your epsilon is how close you want f of x to be to your limit point, right? It's a range around f of x --they'll give you a delta which is a range around a, right? Let me write this. So limit as approaches a of f of x is equal to l. So they'll give you a delta where as long as x is no more than delta-- So the distance between x and a, so if we pick an x here-- let me do another color --if we pick an x here, the distance between that value and a, as long as one, that's greater than 0 so that x doesn't show up on top of a, because its function might be undefined at that point. But as long as the distance between x and a is greater than 0 and less than this x range that they gave you, it's less than delta. So as long as you take an x, you know if I were to zoom the x-axis right here-- this is a and so this distance right here would be delta, and this distance right here would be delta --as long as you pick an x value that falls here-- so as long as you pick that x value or this x value or this x value --as long as you pick one of those x values, I can guarantee you that the distance between your function and the limit point, so the distance between you know, when you take one of these x values and you evaluate f of x at that point, that the distance between that f of x and the limit point is going to be less than the number you gave them. And if you think of, it seems very complicated, and I have mixed feelings about where this is included in most calculus curriculums. It's included in like the, you know, the third week before you even learn derivatives, and it's kind of this very mathy and rigorous thing to think about, and you know, it tends to derail a lot of students and a lot of people I don't think get a lot of the intuition behind it, but it is mathematically rigorous. And I think it is very valuable once you study you know, more advanced calculus or become a math major. But with that said, this does make a lot of sense intuitively, right? Because before we were talking about, look you know, I can get you as close as x approaches this value f of x is going to approach this value. And the way we mathematically define it is, you say Sal, I want to be super close. I want the distance to be f of x [UNINTELLIGIBLE]. And I want it to be 0.000000001, then I can always give you a distance around x where this will be true. And I'm all out of time in this video. In the next video I'll do some examples where I prove the limits, where I prove some limit statements using this definition. And hopefully you know, when we use some tangible numbers, this definition will make a little bit more sense. See you in the next video.

Contents

History

Although the Greeks examined limiting process, such as the Babylonian method, they probably had no concept similar to the modern limit.[3] The need for the concept of a limit arose in the 1600s when Pierre de Fermat attempted to find the slope of the tangent line at a point of a function such as . Using a non-zero, but almost zero quantity, , Fermat performed the following calculation:

The key to the above calculation is that since is non-zero one can divide by , but since is close to 0, is essentially .[4] Quantities such as are called infinitesimals. The problem with this calculation is that mathematicians of the era were unable to rigorously define a quantity with properties of [5] although it was common practice to 'neglect' higher power infinitesimals and this seemed to yield correct results.

This problem reappeared later in the 1600s at the center of the development of calculus because calculations such as Fermat's are important to the calculation of derivatives. Isaac Newton first developed calculus via an infinitesimal quantity called a fluxion. He developed them in reference to the idea of an "infinitely small moment in time..."[6] However, Newton later rejected fluxions in favor of a theory of ratios that is close to the modern definition of the limit.[6] Moreover, Newton was aware that the limit of the ratio of vanishing quantities was not itself a ratio, as he wrote:

Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity...

Additionally, Newton occasionally explained limits in terms similar to the epsilon–delta definition.[7] Gottfried Wilhelm Leibniz developed an infinitesimal of his own and tried to provide it with a rigorous footing, but it was still greeted with unease by some mathematicians and philosophers.[8]

Augustin-Louis Cauchy gave a definition of limit in terms of a more primitive notion he called a variable quantity. He never gave an epsilon–delta definition of limit (Grabiner 1981). Some of Cauchy's proofs contain indications of the epsilon–delta method. Whether or not his foundational approach can be considered a harbinger of Weierstrass's is a subject of scholarly dispute. Grabiner feels that it is, while Schubring (2005) disagrees.[dubious ][1] Nakane concludes that Cauchy and Weierstrass gave the same name to different notions of limit.[9][unreliable source?]

Eventually, Weierstrass and Bolzano are credited with providing a rigorous footing for calculus in the form of the modern definition of the limit. [1][10] The need for reference to an infinitesimal was then removed [11] and Fermat's computation turned into the computation of the following limit:

This is not to say that the limiting definition was free of problems as, although it removed the need for infinitesimals, it did require the construction of the real numbers by Richard Dedekind.[12] This is also not to say that infinitesimals have no place in modern mathematics as later mathematicians were able to rigorously create infinitesimal quantities as part of the hyperreal number or surreal number systems. Moreover, it is possible to rigorously develop calculus with these quantities and they have other mathematical uses.[13]

Informal statement

A viable intuitive or provisional definition is that a "function f approaches the limit L near a (symbolically, ) if we can make f(x) as close as we like to L by requiring that x be sufficiently close to, but unequal to, a."[14]

When we say that two things are close (such as f(x) and L or x and a) we mean that the distance between them is small. When f(x), L, x, and a are real numbers, the distance between two numbers is the absolute value of the difference of the two. Thus, when we say f(x) is close to L we mean is small. When we say that x and a are close, we mean that is small.[15]

When we say that we can make f(x) as close as we like to L, we mean that for all non-zero distances, , we can make the distance between f(x) and L smaller than .[15]

When we say that we can make f(x) as close as we like to L by requiring that x be sufficiently close to, but, unequal to, a, we mean that for every non-zero distance , there is some non-zero distance such that if the distance between x and a is less than then the distance between f(x) and L is smaller than .[15]

The aspect that must be grasped is that the definition requires the following conversation. One is provided with any challenge for a given f,a, and L. One must answer with a such that implies that . If one can provide an answer for any challenge, one has proven that the limit exists.

Precise statement and related statements

Precise statement for real valued functions

The definition of the limit of a function is as follows:[15]

Let be a real-valued function defined on a subset of the real numbers. Let be a limit point of and let be a real number. We say that

if for every there exists a such that, for all , if , then .

Symbolically:

If or , then the condition that is a limit point is automatically met because closed real intervals and the entire real line are perfect sets.

Precise statement for functions between metric spaces

The definition can be generalized to functions that map between metric spaces. These spaces come with a function, called a metric, that takes two points in the space and returns a real number that represents the distance between the two points.[16] The generalized definition is as follows:[17]

Suppose is defined on a subset of a metric space with a metric and maps into a metric space with a metric . Let be a limit point of and let be a point of .

We say that

if for every there exists a such that, for all , if , then .

Since is a metric on the real numbers, one can show that this definition generalizes the first definition for real functions.[18]

Negation of the precise statement

The negation of the definition is as follows:[19]

Suppose is defined on a subset of a metric space with a metric and maps into a metric space with a metric . Let be a limit point of and let be a point of .

We say that

if there exists an such that for all there is an such that and .

We say that does not exist if for all , .

For the negation of a real valued function defined on the real numbers, simply set .

Precise statement for limits at infinity

The precise statement for limits at infinity is as follows:[16]

Suppose is defined on a subset of a metric space with a metric and maps into a metric space with a metric . Let .

We say that

if for every , there is a real number such that there is an where and such that if and , then .

Worked examples

Example 1

We will show that

.

We let be given. We need to find a such that implies .

Since sine is bounded above by 1 and below by -1,

Thus, if we take , then implies , which completes the proof.

Example 2

Let us prove the statement that

for any real number .

Let be given. We will find a such that implies .

We start by factoring:

We recognize that is the term bounded by so we can presuppose a bound of 1 and later pick something smaller than that for .[20]

So we suppose . Since holds in general for real numbers and , we have

Thus,

Thus via the triangle inequality,

Thus, if we further suppose that

then

.

In summary, we set

.

So, if , then

Thus, we have found a such that implies . Thus, we have shown that

for any real number .

Example 3

Let us prove the statement that

This is easily shown through graphical understandings of the limit, and as such serves as a strong basis for introduction to proof. According to the formal definition above, a limit statement is correct if and only if confining to units of will inevitably confine to units of . In this specific case, this means that the statement is true if and only if confining to units of 5 will inevitably confine

to units of 12. The overall key to showing this implication is to demonstrate how and must be related to each other such that the implication holds. Mathematically, we want to show that

Simplifying, factoring, and dividing 3 on the right hand side of the implication yields

which immediately gives the required result if we choose

Thus the proof is completed. The key to the proof lies in the ability of one to choose boundaries in , and then conclude corresponding boundaries in , which in this case were related by a factor of 3, which is entirely due to the slope of 3 in the line

Continuity

A function f is said to be continuous at c if it is both defined at c and its value at c equals the limit of f as x approaches c:

The definition for a continuous function can be obtained from the definition of a limit by replacing with to ensure that f is defined at c and equals the limit.

f is said to be continuous on an interval I if it is continuous at every point c of I.

Comparison with infinitesimal definition

Keisler proved that a hyperreal definition of limit reduces the quantifier complexity by two quantifiers.[21] Namely, converges to a limit L as tends to a if and only if for every infinitesimal e, the value is infinitely close to L; see microcontinuity for a related definition of continuity, essentially due to Cauchy. Infinitesimal calculus textbooks based on Robinson's approach provide definitions of continuity, derivative, and integral at standard points in terms of infinitesimals. Once notions such as continuity have been thoroughly explained via the approach using microcontinuity, the epsilon–delta approach is presented as well. Karel Hrbáček argues that the definitions of continuity, derivative, and integration in Robinson-style non-standard analysis must be grounded in the ε–δ method in order to cover also non-standard values of the input.[22] Błaszczyk et al. argue that microcontinuity is useful in developing a transparent definition of uniform continuity, and characterize the criticism by Hrbáček as a "dubious lament".[23] Hrbáček proposes an alternative non-standard analysis, which (unlike Robinson's) has many "levels" of infinitesimals, so that limits at one level can be defined in terms of infinitesimals at the next level.[24]

See also

References

  1. ^ a b c Grabiner, Judith V. (March 1983), "Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus" (PDF), The American Mathematical Monthly, 90 (3): 185–194, doi:10.2307/2975545, JSTOR 2975545, archived (PDF) from the original on 2009-05-04, retrieved 2009-05-01
  2. ^ Cauchy, A.-L. (1823), "Septième Leçon - Valeurs de quelques expressions qui se présentent sous les formes indéterminées  Relation qui existe entre le rapport aux différences finies et la fonction dérivée", Résumé des leçons données à l'école royale polytechnique sur le calcul infinitésimal, Paris, archived from the original on 2009-05-04, retrieved 2009-05-01, p. 44.. Accessed 2009-05-01.
  3. ^ Stillwell, John (1989). Mathematics and its history. New York: Springer-Verlag. pp. 38–39. ISBN 978-1-4899-0007-4.
  4. ^ Stillwell, John (1989). Mathematics and its history. New York: Springer-Verlag. p. 104. ISBN 978-1-4899-0007-4.
  5. ^ Stillwell, John (1989). Mathematics and its history. New York: Springer-Verlag. p. 106. ISBN 978-1-4899-0007-4.
  6. ^ a b Buckley, Benjamin Lee (2012). The continuity debate : Dedekind, Cantor, du Bois-Reymond and Peirce on continuity and infinitesimals. p. 31. ISBN 9780983700487.
  7. ^ Pourciau, B. (2001), "Newton and the Notion of Limit", Historia Mathematica, 28 (1): 18–30, doi:10.1006/hmat.2000.2301
  8. ^ Buckley, Benjamin Lee (2012). The continuity debate : Dedekind, Cantor, du Bois-Reymond and Peirce on continuity and infinitesimals. p. 32. ISBN 9780983700487.
  9. ^ Nakane, Michiyo. Did Weierstrass's differential calculus have a limit-avoiding character? His definition of a limit in ε−δ style. BSHM Bull. 29 (2014), no. 1, 51–59.
  10. ^ Cauchy, A.-L. (1823), "Septième Leçon - Valeurs de quelques expressions qui se présentent sous les formes indéterminées  Relation qui existe entre le rapport aux différences finies et la fonction dérivée", Résumé des leçons données à l'école royale polytechnique sur le calcul infinitésimal, Paris, archived from the original on 2009-05-04, retrieved 2009-05-01, p. 44..
  11. ^ Buckley, Benjamin Lee (2012). The continuity debate : Dedekind, Cantor, du Bois-Reymond and Peirce on continuity and infinitesimals. p. 33. ISBN 9780983700487.
  12. ^ Buckley, Benjamin Lee (2012). The continuity debate : Dedekind, Cantor, du Bois-Reymond and Peirce on continuity and infinitesimals. pp. 32–35. ISBN 9780983700487.
  13. ^ Tao, Terence (2008). Structure and randomness : pages from year one of a mathematical blog. Providence, R.I.: American Mathematical Society. pp. 95–110. ISBN 978-0-8218-4695-7.
  14. ^ Spivak, Michael (2008). Calculus (4th ed.). Houston, Tex.: Publish or Perish. p. 90. ISBN 978-0914098911.
  15. ^ a b c d Spivak, Michael (2008). Calculus (4th ed.). Houston, Tex.: Publish or Perish. p. 96. ISBN 978-0914098911.
  16. ^ a b Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill Science/Engineering/Math. p. 30. ISBN 978-0070542358.
  17. ^ Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill Science/Engineering/Math. p. 83. ISBN 978-0070542358.
  18. ^ Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill Science/Engineering/Math. p. 84. ISBN 978-0070542358.
  19. ^ Spivak, Michael (2008). Calculus (4th ed.). Houston, Tex.: Publish or Perish. p. 97. ISBN 978-0914098911.
  20. ^ Spivak, Michael (2008). Calculus (4th ed.). Houston, Tex.: Publish or Perish. p. 95. ISBN 978-0914098911.
  21. ^ Keisler, H. Jerome (2008), "Quantifiers in limits" (PDF), Andrzej Mostowski and foundational studies, IOS, Amsterdam, pp. 151–170
  22. ^ Hrbacek, K. (2007), "Stratified Analysis?", in Van Den Berg, I.; Neves, V. (eds.), The Strength of Nonstandard Analysis, Springer
  23. ^ Błaszczyk, Piotr; Katz, Mikhail; Sherry, David (2012), "Ten misconceptions from the history of analysis and their debunking", Foundations of Science, 18: 43–74, arXiv:1202.4153, doi:10.1007/s10699-012-9285-8
  24. ^ Hrbacek, K. (2009). "Relative set theory: Internal view". Journal of Logic and Analysis. 1.

Further reading

This page was last edited on 8 August 2019, at 08:25
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