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This thread was and still is in the topology forum, not in a homework forum. It is labeled "I", not "B". I think that Weierstraß can be discussed here and "freshman" is not warranted.FactChecker said:
##\epsilon - \delta## proof and algebraic proof of limits
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In summary, the conversation discusses finding the limit of a sequence, specifically ##s_n = \frac{3n+1}{7n-4}##. The speaker presents two approaches, one involving intuition and the other using an ##\epsilon-\delta## proof. The first attempt is considered more rigorous as it presents a more concrete understanding of the limit, while the second attempt has a flaw in its logic. The speaker also mentions the importance of the ##\epsilon-\delta## proof as the gold standard, even though it may not always be as obvious as in this specific example.
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Mark44
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That's Topology and Analysis. You can't always trust that a thread starter picks the best forum for a question.fresh_42 said:
"I" is college level, which includes the freshman class. Taking another look at the first post in this thread, my guess is that the OP is taking freshman-level calculus.fresh_42 said:
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Even if. The Weierstraß definition is not over a freshman's head. I think it's even easier, but definitely not harder.Mark44 said:
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FactChecker
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I am afraid that I caused a digression from the OP to derivatives.fresh_42 said:
In any case, we have presented the OP with alternatives. I think that he can see what is involved in each approach and decide what is most appropriate for him. I would caution him this way:
1) The ##\epsilon, \delta## approach stays within very basic arithmetic with very little prior development needed.
2) The Weierstrass approach is for calculus and is a good approach for more generalized (multidimensional, directional, etc.) derivative definitions.
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Hall
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@FactChecker is right, epsilon-delta proofs are easier.
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Hall
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Weierstrass decomposibility as an alternative to differentiability is, of course, not over my head, but I really cannot say anything about Freshmen. But the problem with Weierstrass decomposability is that I couldn't find source of it, nothing was available of it (given my hard work and Google fu skills). The English translation of the link given by @fresh_42 explained very little, but all I got was: a function at point ##x_0## can be decomposed as the value of the tangent line at that point plus an error function.
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Try to view it from a physicist's point of view. A derivative in physics is a linear function in the first place, not a number. The approach via the limit of a slope yields a number, not a function. In Weierstraß's formula, the linear function is the main part of it; later specified as gradient or Jacobi matrix.Hall said:
If you learn (at school) that the derivative gives a slope, and learn next that a derivative gives a gradient (first semester), then that the gradient is a linear function (second semester), and finally, that it is a section of a cotangent bundle (sixth semester), then you will have learned the same thing four times. Maybe it is a bit exaggerated to start with vector bundles, but Weierstraß's formula reduces the other three to one lesson and an easy proof that both are equivalent.
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You might not see it referenced by name, but you might see it used as the definition of the derivative in higher dimensions. As @fresh_42 says, it is the way that many (most?) people like to think of the derivative and visualize it, especially in higher dimensions. It immediately makes a lot of matrix theory and linear algebra apply straight from the definition. A person who has already spent years learning linear operators would find it painful to go back to an ##\epsilon, \delta## definition.Hall said:
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Mark44
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The limit of the slope at a particular point on the graph of a function yields a number, but if you do the calculation at an arbitrary point, you get a function.fresh_42 said:
Suppose that ##f(x) = x^2##.
$$f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}h = \lim_{h \to 0}\frac{(x + h)^2 - x^2} h = \lim_{h \to 0} \frac{2xh + h^2}h$$
$$= \lim_{h \to 0} \frac {h(2x + h)}h = \lim_{h \to 0} \frac h h \cdot \lim_{h \to 0}2xh = 2x$$
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This is another reason why I do not like the limit definition. Of course, you are right and this is a function depending on the variable location. Make a survey and ask how many students are aware of the fact that the evaluation point is the variable! And how is ##g(x):=x^3\, , \,g'(x)=3x^2## a linear function?Mark44 said:
In reality, it is far too often used when actually ##\left. \dfrac{d}{dx}\right|_{x=x_0}f## has been meant.
We can discuss this forever, and I know I'm fighting windmills. But being a minority does not change my opinion. Here is a list of ten meanings for the derivative and the word slope wasn't even among them:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/
This is why I am of the opinion: Teach right not twice.
- #46
Mark44
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I don't need to make a survey, but I have taught calculus more times than I can recall. Students who received a passing grade in the class knew that the variable was the point at which the derivative was calculated.fresh_42 said:
fresh_42 said:
Well, that's the difference between ##f'(x_0)## and ##f'(x)##. Again, this is stuff I tested my students on.
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Look at my example, ##g(x)=x^3.## Then Weierstaß's formula reads
$$
g(x_0+v)=(x_0+v)^3=\underbrace{x_0^3}_{=g(x_0)}+\underbrace{\underbrace{3x_0^2}_{=g'(x_0)}\cdot \underbrace{v}_{\text{direction}}}_{\text{slope}}+\underbrace{v^2(3x_0+v)}_{=r(v)}
$$
I simply think that this explains all relevant parts better than a limit, especially the fact that all derivatives are directional. But do not misunderstand me. I still think that the limit definition has its merits. It is just very easy to teach both and use whatever fits best in a specific situation.
$$
g(x_0+v)=(x_0+v)^3=\underbrace{x_0^3}_{=g(x_0)}+\underbrace{\underbrace{3x_0^2}_{=g'(x_0)}\cdot \underbrace{v}_{\text{direction}}}_{\text{slope}}+\underbrace{v^2(3x_0+v)}_{=r(v)}
$$
I simply think that this explains all relevant parts better than a limit, especially the fact that all derivatives are directional. But do not misunderstand me. I still think that the limit definition has its merits. It is just very easy to teach both and use whatever fits best in a specific situation.
Last edited:
- #48
nuuskur
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You could prove in one fell swoop that for polynomials ##f,g## of equal degree and with leading coefficients ##a,b## respectively,
[tex]
\lim _{|x|\to\infty} \frac{f(x)}{g(x)} = \frac{a}{b}
[/tex]
using epsilon delta trickery and never have to worry about specific cases again
[tex]
\lim _{|x|\to\infty} \frac{f(x)}{g(x)} = \frac{a}{b}
[/tex]
using epsilon delta trickery and never have to worry about specific cases again
- #49
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Interestingly, that's another intuitive approach to the relation ## \delta - \epsilon ##, which is similar to ## \lim _{ \Delta x \rightarrow 0} \frac { \Delta f}{ \Delta x} ##FactChecker said:
<h2>1. What is the purpose of an ##\epsilon - \delta## proof?</h2><p>The purpose of an ##\epsilon - \delta## proof is to rigorously prove the existence of a limit by showing that for any small distance ##\epsilon##, there exists a corresponding small distance ##\delta## such that if the input to the function is within ##\delta## of the limit, the output will be within ##\epsilon## of the limit.</p><h2>2. How is an ##\epsilon - \delta## proof different from an algebraic proof of limits?</h2><p>An ##\epsilon - \delta## proof is a more rigorous and precise method of proving the existence of a limit, while an algebraic proof relies on manipulating and simplifying the limit expression to show that it approaches a certain value. ##\epsilon - \delta## proofs are often used in calculus courses, while algebraic proofs may be used in more advanced mathematics courses.</p><h2>3. What is the role of the ##\epsilon## and ##\delta## values in an ##\epsilon - \delta## proof?</h2><p>The ##\epsilon## and ##\delta## values represent small distances that are used to define the closeness of the input and output values to the limit. The goal of an ##\epsilon - \delta## proof is to show that for any small distance ##\epsilon##, there exists a corresponding small distance ##\delta## such that the input and output values are within ##\epsilon## of the limit.</p><h2>4. Can an ##\epsilon - \delta## proof be used to prove the non-existence of a limit?</h2><p>No, an ##\epsilon - \delta## proof can only be used to prove the existence of a limit. If an ##\epsilon - \delta## proof fails, it does not necessarily mean that the limit does not exist. It may just mean that a different approach is needed to prove the existence of the limit.</p><h2>5. What are the advantages of using an ##\epsilon - \delta## proof over other methods of proving limits?</h2><p>An ##\epsilon - \delta## proof provides a more rigorous and precise approach to proving limits, as it relies on the definition of a limit rather than manipulation of expressions. It also allows for a better understanding of the behavior of a function near a limit point. Additionally, ##\epsilon - \delta## proofs can be used to prove the existence of limits for more complex functions that may not have algebraic solutions.</p>
1. What is the purpose of an ##\epsilon - \delta## proof?
The purpose of an ##\epsilon - \delta## proof is to rigorously prove the existence of a limit by showing that for any small distance ##\epsilon##, there exists a corresponding small distance ##\delta## such that if the input to the function is within ##\delta## of the limit, the output will be within ##\epsilon## of the limit.
2. How is an ##\epsilon - \delta## proof different from an algebraic proof of limits?
An ##\epsilon - \delta## proof is a more rigorous and precise method of proving the existence of a limit, while an algebraic proof relies on manipulating and simplifying the limit expression to show that it approaches a certain value. ##\epsilon - \delta## proofs are often used in calculus courses, while algebraic proofs may be used in more advanced mathematics courses.
3. What is the role of the ##\epsilon## and ##\delta## values in an ##\epsilon - \delta## proof?
The ##\epsilon## and ##\delta## values represent small distances that are used to define the closeness of the input and output values to the limit. The goal of an ##\epsilon - \delta## proof is to show that for any small distance ##\epsilon##, there exists a corresponding small distance ##\delta## such that the input and output values are within ##\epsilon## of the limit.
4. Can an ##\epsilon - \delta## proof be used to prove the non-existence of a limit?
No, an ##\epsilon - \delta## proof can only be used to prove the existence of a limit. If an ##\epsilon - \delta## proof fails, it does not necessarily mean that the limit does not exist. It may just mean that a different approach is needed to prove the existence of the limit.
5. What are the advantages of using an ##\epsilon - \delta## proof over other methods of proving limits?
An ##\epsilon - \delta## proof provides a more rigorous and precise approach to proving limits, as it relies on the definition of a limit rather than manipulation of expressions. It also allows for a better understanding of the behavior of a function near a limit point. Additionally, ##\epsilon - \delta## proofs can be used to prove the existence of limits for more complex functions that may not have algebraic solutions.
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