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29 votes
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Near-universal student mistake on $\lim_{x\rightarrow\infty}e^{x+1}/e^x$

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 ...
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23 votes
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Should we tell students to never replace parts of an expression by their limits when taking a limit?

The important thing is whether students' reasoning is logically valid β€” and in particular, that they only use the conclusion of a theorem after they've checked that all its hypotheses hold β€” not ...
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  • 4,765
20 votes

What is the most difficult concept to grasp in Calculus 1?

At my (community) college, I believe many teachers skip the (πœ–,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 ...
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20 votes
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How should students say in words the notation for a limit?

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 ...
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  • 2,008
16 votes

Near-universal student mistake on $\lim_{x\rightarrow\infty}e^{x+1}/e^x$

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 ...
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  • 261
14 votes

Nice examples of limits to infinity in real life

If you are willing to take some time to explain the model and do some simulation, I really like to show students a logistic growth model (in discrete time). The basic setup is something like the ...
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10 votes
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What's the best way to explain multivariable limit problems to students who are not familiar with $\epsilon-\delta$ proofs?

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 ...
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  • 5,752
10 votes

What's the best way to explain multivariable limit problems to students who are not familiar with $\epsilon-\delta$ proofs?

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 ...
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  • 4,784
9 votes
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What is the intuition behind the limit superior?

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 ...
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  • 2,932
9 votes

When should we get into limits in introductory calculus courses?

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 ...
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9 votes

Any metaphors/intuitions for a limit of a sequence?

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 ...
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9 votes

When should we get into limits in introductory calculus courses?

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 ...
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9 votes
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When evaluating the limit of $f(x, y)$ as $(x, y)$ approaches $(x_0, y_0)$, should we consider only those $(x, y)$ in the domain of $f$?

Most definitions of a limit of a function only include the domain. See the Wikipedia article about limits or this textbook about Real Analysis (or the various textbooks in this Math StackExchange ...
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  • 402
8 votes

How long would it take to teach proper limit calculations?

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 ...
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  • 13.5k
8 votes

How should students say in words the notation for a limit?

I say it the third way, for these reasons: Firstly, from a notation point of view, the β€œ$x\to a$” has to be written with the β€œ$\lim$”, and no β€œ$\lim$” can be written without it (without specifically ...
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8 votes

An intuitive explanation of l'Hôpital's rule for ∞/∞

Your approach is in fact geometrical, if we see all this happening on the Riemann sphere, i.e., the one-point compactification of ${\bf C}={\bf R}^2$. Briefly, we have the usual coordinate $z$ on ${\...
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  • 7,575
7 votes

What is the most difficult concept to grasp in Calculus 1?

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 ...
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  • 12k
7 votes

Nice examples of limits to infinity in real life

Deeba and Rushkady Go to Town: A Fanciful Real-Life Story A festival was just starting in town, and Deeba and Rushkady walked toward the square headed to the Infinite Pancake Eating Contest. The ...
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  • 4,579
6 votes

Should we tell students to never replace parts of an expression by their limits when taking a limit?

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} (...
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6 votes

How can I explain $\lim_{x \to \infty} \frac{e^x+e^{-x}}{e^x-e^{-x}}$ using L'Hôpital's Rule?

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 ...
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  • 6,211
6 votes

How long would it take to teach proper limit calculations?

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 ...
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  • 617
6 votes

How can we explain and justify different results of universal quantification?

The logical and consistent account is to actually use logic. The limit has nothing to do with the behaviour 'before' that. When you go to the mall, you are "on the way" all the way until you reach, at ...
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  • 2,261
6 votes
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Interesting but very easy epsilon-delta problems?

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 ...
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6 votes

How to correct a wrong mental picture of the limit?

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 ...
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  • 4,784
6 votes

How to correct a wrong mental picture of the limit?

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). ...
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  • 16.3k
6 votes

When teaching someone how to prove a function is uniformly continuous, using epsilon/delta, which example would be among the simplest?

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 ...
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6 votes
Accepted

Real World use of the Function $(\sin{x})^x$

I suspect that if $(\sin x)^x$ shows up in any physical situation, it will be highly specific and not really a natural or worthwhile thing for a calculus student to spend their time on. Perhaps ...
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  • 4,784
6 votes

What is the most difficult concept to grasp in Calculus 1?

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 ...
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  • 114
5 votes

Teaching limits and asymptotics at the same time

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 ...
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  • 2,496

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