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27

While all of your students at this point will have done (extensive) units on manipulating and simplifying expressions with exponents, this is the limits unit. When doing limits questions, most students are only searching their brains for limits techniques - they're extremely unlikely to come up with the $\frac{x^a}{x^b} = x^{a-b}$ rule that had been driven ...


22

The important thing is whether students' reasoning is logically valid — and in particular, that they only use the conclusion of a theorem after they've checked that all its hypotheses hold — not whether they follow any particular arbitrary rules or procedures. In this case, the relevant theorem is the following: Theorem. Let $f$ and $g$ be real-valued ...


19

At my (community) college, I believe many teachers skip the (𝜖,𝛿)-definitions of limits. I give the main definition, and teach it with an example I call the cookie crispness index. But I don't assign problems in that section, and I don't test on it. I figure actually working with it is for an analysis course. Of the topics that I do test on, implicit ...


15

First of all, I think that the problem statement can be confusing. use L'Hospital's rule if possible, or if not, explain why it didn't work and evaluate it by some other method It can be misleading for students because it is easy to assume that if a fairly complicated method does not work (L'Hospital's rule), what you should do is use an even more ...


13

According to UCAR-10101: Handbook for Spoken Mathematics, Lawrence A. Chang, Ph.D., page 38, a source for how to speak mathematics to sight-impaired students, we have: $$\lim_{x\to a} y = b$$ is spoken as the "limit as $x$ approaches $a$ of $y$ equals $b$". For the given expression, $$\lim_{x\to a} f(x) = L$$ is spoken thusly: the "limit as $x$ approaches $...


11

I'd like to know why textbooks cover limits first. I don't think there's any big mystery as to why commercial textbooks tend to be similar. It's a market mechanism known as the network effect, the same mechanism that makes Microsoft Windows so popular. Once people start to see something as a standard, anything different becomes non-viable in the marketplace....


10

Showing why there is a need for good definitions here is probably more important than showing students why a difficult definition "works" via plug-in hocus-pocus. (Though if you have the time to do the proper definition properly that is even better; I certainly don't in multivariate calc. Squeeze theorem, as commenter John Coleman points out, is a good ...


10

You can make this $\epsilon/\delta$ proof easy enough that an interested student should find the argument believable without experience with $\epsilon/\delta$ proofs. Divide both the numerator and denominator by $x^2 y^2 z^2$ to get $$\frac{1}{ \frac{1}{y^2 z^2} + \frac{1}{x^2 z^2} + \frac{1}{y^2 z^2}}$$ Then if $|x|,|y|,|z| < \epsilon$ then we have $$ \...


9

My currently preferred approach is to start the course with an introductory lecture explaining the difference between average velocity over a time interval (something we can always in principle measure using a stopwatch) and instantaneous velocity at a given instant of time (which we do not have a means to measure). This motivates the question of what ...


9

My feeling is that the $\epsilon$-$\delta$ formulation is already pretty close to what one should think about limits; that is, the language can be hard to grasp at first, but the idea is very literally expressed in that language and can also be expressed in words. I find this more convincing using a physics measurement approach (which is slightly more ...


9

It has become almost a dogma that the math curriculum should teach technical prerequisites to what will be covered later. The consequence is that zillions of high-school students learn algorithms for doing partial-fractions decompositions and will never take later courses in which that is used. In effect the broad public learns that mathematics consists of ...


8

I think it is too hasty to dismiss manipulation of "$\infty$" out-of-hand, although, yes, there is a widespread tendency among students to over-simplify, thus crossing various lines into trouble. The first-presented version is not so bad as a heuristic, and might have been written by Euler or Lagrange. For that matter, it can be made more focused by $\lim{x-...


8

I have two intuitions to offer: A sequence $(a_n)$ may have cluster points (these are points such that every neighborhood contains infinitely many elements of the sequence, or, more precisely, for every neighborhood and every index $N$, there is an element $a_n$ with an index $n>N$ which is in this neighborhood). The $\limsup$ is the largest of these ...


8

I say it the third way, for these reasons: Firstly, from a notation point of view, the “$x\to a$” has to be written with the “$\lim$”, and no “$\lim$” can be written without it (without specifically saying what you mean by not having it), so it makes sense to put them together when you say it aloud. Indeed, you could argue that $\displaystyle\lim_{x \to a}...


7

I'd say the whole $\epsilon/\delta$ dance. Just because it involves inequalities, while students just have trained equalities all their life. To just work with (sometimes very rough) bounds, when you were drilled to get exact answers goes against the grain.


6

To extend the answer by Daniel Hast: One theorem one might want to use is: If $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ are convergent sequences then $$\begin{align} \lim_{n\to\infty} (a_n \pm b_n) &= \lim_{n\to\infty} a_n \pm \lim_{n\to\infty} b_n \\ \lim_{n\to\infty} (a_n \cdot b_n) &= \lim_{n\to\infty} a_n \cdot \lim_{n\to\infty} ...


6

As noted, l'Hopital directly fails. So you have to transform somehow first. For example, $$ \frac{e^x+e^{-x}}{e^x-e^{-x}} = \frac{e^{2x}+1}{e^{2x}-1} $$ then l'Hopital works. But of course the same method $$ \frac{e^x+e^{-x}}{e^x-e^{-x}} = \frac{1+e^{-2x}}{1-e^{-2x}} \to \frac{1+0}{1-0} $$ solves it without l'Hopital at all.


6

It might be enough to show an example or two of what can go wrong with the logic of the first solution: $\lim_{x\to\infty} \frac{e^x}{x^2}=\frac{\infty}{\infty^2}=\frac{1}{\infty}=0$ But of course this is incorrect! Here's one more, in case they don't know the exponential-polynomial comparison: $\lim_{x\to\infty}\frac{x^2}{\sqrt{x+1}\sqrt{x+2}\sqrt{x+3}}=...


6

If your students really think that the limit is something that is realized after infinitely many steps, I think that you are doing them a disservice trying to dissuade them. Taking the limit is effectively a transfinite operation which does just that. (My students sometimes seem to get the impression that you evaluate finitely many terms then round, ...


6

I am genuinely surprised that in a SE site populated by (among others) math education researchers, nobody has yet mentioned Tall & Vinner's landmark paper: Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational studies in mathematics, 12(2), 151-169. (...


6

I think this cannot be understood without a contrasting example where it fails. So perhaps, in addition to a linear function as suggested by @paw88789, consider $f(x) = \frac{1}{x}$ over the open interval $(0,1)$. It is continuous over that interval, but not uniformly continuous. Fix an $\epsilon > 0$; then for any $\delta > 0$ one can arrange the ...


6

Related rates problems show the most issues on AP grading. (No reference, just my own observation) Topic involves word problems, geometry and some multistep problem solving. I don't think the concept of epsilon delta is so difficult as the tediousness of the algebra along with students openly or unconsciously questioning the value add for science and ...


5

If I understand you, I do often think this way when solving a problem. For example, see this answer of mine at Mathematics Stack Exchange. However, limits involving infinity (as the independent or dependent variable) seem to be losing importance in textbooks. For example, I teach from Calculus: Graphical, Numerical, Algebraic by Ross L. Finney et al., and ...


5

Perhaps this is not a good answer per-se, insofar as it yet-once-again questions (to a certain degree) the premises of the question... But, indeed, in many propitious situations, everything in sight is continuous, and it is justifiable to replace sub-expressions with their limits. I might claim that the practical success of calculus substantially depends ...


5

I suggest using rational functions. Students are used to evaluating limits of rational functions because such examples are prevalent in most calculus courses. Moreover, I think the work required to prove such a limit is a little more challenging than that for a linear function but mostly in terms of the algebra and a little bit of logic. But, by following ...


4

Regarding your parenthetical comment "And it turns out that limits are not the only way to do so. Non-standard analysis uses infinitesimals in a logically rigorous way" I would like to comment that the opposition limits vs infinitesimals implied here is not entirely accurate. Limits are present in both approaches; the true opposition is between epsilon-...


4

I'd recommend not being intimidated by textbook trends! "In real life" (as opposed to textbook-life, for sure, and often opposed to required-curriculum "school-math"), _of_course_ the two things you mention, "the limit", and "how it is approached", matter a great deal. Do not be intimidated by silly books (written by non-mathematicians, almost entirely) to ...


4

The idea behind the limsup that you write is not simple and will not convey a concise intuition. The shortest description of the limsup of a sequence needs two steps: (1) The audience has to know what a limit point (also called accumulation point or cluster point) of a sequence is, and that a sequence can have many limit points, by examples. (2) Granting ...


4

$\displaystyle \lim_{(x,y,z)\rightarrow(0,0,0)} \frac{x^2y^2z^2}{x^2+y^2+z^2}$ Even if this were a course where students were required to write epsilon-delta proofs, I would still imagine that anyone competent would approach this problem by first thinking informally about why this limit is expected to exist and be zero. I would split this into three levels ...


4

Welcome to SE. As you are asking on a side for math educators, I assume that you want to explain this to students. My advice: don't. Start with much, much easier examples to get the students accustomed to such problems, then come back to it later; if at all. If you, as the teacher/professor, are not able to find a solution for this problem, it is too hard ...


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