# Tag Info

### "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 $\theta \approx 0$) ...
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### 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 ...
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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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### "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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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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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### 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 $\mathbb{R}^n$, ...
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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 ...
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### 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 ...
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