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32 votes

"Real life" examples of limits of functions at finite points

First thing that comes to my mind is the limit $$\lim\limits_{x \to 0} \dfrac{\sin x}{x} = 1.$$ This limit justifies the small-angle approximation $\sin \theta \approx \theta$ (for $\theta \approx 0$) ...
Justin Skycak's user avatar
21 votes

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

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 ...
Sue VanHattum's user avatar
  • 21k
20 votes
Accepted

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 ...
amWhy's user avatar
  • 2,095
20 votes

"Real life" examples of limits of functions at finite points

"interesting, natural and simple" Illustrating something dynamically might make things interesting. For example, a geometry problem involving a limit (from an old calculus book): Consider a ...
Nick C's user avatar
  • 9,719
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 ...
Xander Henderson's user avatar
  • 8,263
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 ...
Adam's user avatar
  • 5,873
10 votes
Accepted

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 ...
kcrisman's user avatar
  • 5,986
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 ...
Michael Hardy's user avatar
9 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 “$x\to a$” has to be written with the “$\lim$”, and no “$\lim$” can be written without it (without specifically ...
DavidButlerUofA's user avatar
9 votes
Accepted

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 ...
Reed Oei's user avatar
  • 412
9 votes

Is this motivation for the concept of a limit a good one?

The concept of a limit has nothing to do with the order on $\mathbb{R}$. The standard definition of a limit of a sequence generalizes, almost verbatim, to sequences with values in $\mathbb{R}^n$, ...
Kostya_I's user avatar
  • 1,411
9 votes
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Process of finding limits for multivariable functions

There's nothing wrong with "changing" or substituting expressions for both $x$ and $y$. However, we can only use one variable across both expressions. What does looking at a function along ...
Justin Hancock's user avatar
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 ${\...
user52817's user avatar
  • 11k
7 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 ...
Adam's user avatar
  • 5,873
7 votes

About the word "limit" used in calculus

One of the difficult aspect of learning and teaching mathematics is that it is in part akin to learning or teaching a new language. Words are used with precise technical meanings that are often ...
Dan Fox's user avatar
  • 5,869
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 ...
vonbrand's user avatar
  • 12.3k
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 ...
user1815's user avatar
  • 5,760
7 votes

Is this motivation for the concept of a limit a good one?

I don't like it. Had a hard time following it. Just tuned out. Yes, I'm not a Ph.D. in math. But neither will be the target students. You should have won me over. You didn't. I have the IQ to ...
guest's user avatar
  • 199
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 ...
guest2's user avatar
  • 114
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). ...
mweiss's user avatar
  • 17.4k
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 ...
Adam's user avatar
  • 5,873
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 ...
Joseph O'Rourke's user avatar
6 votes
Accepted

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 ...
Brendan W. Sullivan's user avatar
5 votes

Frequent calculus error: replacing interior part of an expression with its limit

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 ...
paul garrett's user avatar
  • 14.7k
5 votes

Terminology for parts of limit notation

I vote for Pedro. We should call the point at which the limit is taken the limit point. In contrast, the value obtained by the limit (if it exists) is the limit's value. In particular, $$ \lim_{x \...
James S. Cook's user avatar
5 votes

Calculus limits taught in the US vs Spain?

IME, it's a generational thing. When I was in HS in the US in the early 80's, we studied convergent sequences and Cauchy sequences in the year before AP Calculus. (We didn't get to the punchline ...
Matthew Daly's user avatar
  • 5,629
5 votes
Accepted

Finite sum of infinite series

"Infinite sum" is in common use, so it should be acceptable to say that $2$ is the value of the infinite sum $1+\frac{1}{2}+\frac{1}{4}+\ldots$. But students need to be very clear that the ...
Will Orrick's user avatar
  • 1,122
4 votes

When should we get into limits in introductory calculus courses?

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 ...
Mikhail Katz's user avatar
  • 2,238
4 votes
Accepted

Multivariable limit problem

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 ...
Dirk's user avatar
  • 1,318

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