# 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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### 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 ...

### 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 ...

### "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 ...

### 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 ...

### 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 ...

### 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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### 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 ...

### 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 ...

### 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 ...

### 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 ...

### 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 ...