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Questions tagged [limits]

For question regarding the properties and evaluation of limits.

1
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1answer
135 views

About the word “limit” used in calculus

In the introduction of the limits chapter of a scholar book I can read (translated and abstract) the following examples: "the price of a product has a limit value that, from this price, the number of ...
-4
votes
1answer
160 views

When should the limit be introduced?

Usually school children are taught fractions and decimal representations way before the notion of limit. So they must come to the idea that infinite decimal sequences like 0.999... are the same as an ...
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6answers
417 views

How to correct a wrong mental picture of the limit?

According to my experience many students get in school a wrong mental picture of the limit as something that is realized after infinitely many steps. They think that $0.999...$ and $\sum\limits_{n=1}^\...
7
votes
3answers
423 views

Interesting but very easy epsilon-delta problems?

I am teaching a real analysis class. Students in the class have inconsistent high school algebra skills. They now have a complete but tenuous understanding of $\varepsilon$-$\delta$ limits. I want to ...
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votes
2answers
312 views

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

I will give some examples of universal quantification where the "for all" aspect holds or is violated in the limit. In some examples a failure is preserved, in others it is not. (1) $\forall$ $n \in \...
0
votes
1answer
64 views

Multivariable limit problem [closed]

Im triying to explain this delta-epsilon problem, but I didnt find a way to attack effectively this rigorous demonstration I actually i tried a lot of inequalities (Cauchy-Schwarz etc), but nothing ...
8
votes
4answers
252 views

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

For example, $\displaystyle \lim_{(x,y,z)\rightarrow(0,0,0)} \frac{x^2y^2z^2}{x^2+y^2+z^2}$ This question is from 8th edition of Stewart Calculus textbook. My fellow graduate student TAs and ...
16
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5answers
284 views

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

For example $$\lim\limits_{n\to\infty}\left(1+\frac{1}{2n+1}\right)^{n} =\lim\limits_{n\to\infty}{1}^{n}=1\,.$$ Here the student has replaced the sub-part $\frac{1}{2n+1}$ with its limit $0$, but he ...
5
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4answers
503 views

What is the intuition behind the limit superior?

I want to write an article which explains the limit superior. I also want to present the intuition behind this concept. Currently I would describe the limit superior as the "least upper bound of a ...
2
votes
1answer
205 views

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

I have given a problem in limits to my students: $$\lim_{x \to \infty} \frac{e^x+e^{-x}}{e^x-e^{-x}}$$ Most of the students used direct substitution and identified that it is an indeterminate form $...
29
votes
3answers
954 views

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

On a recent first-semester calculus exam, I gave a bunch of limits. The student was supposed to use L'Hospital's rule if possible, or if not, explain why it didn't work and evaluate it by some other ...
8
votes
5answers
449 views

How long would it take to teach proper limit calculations?

This question arose from discussion of this question. How long would it take you to teach typical undergratuate (calculus) students the difference between the following two calculations? $$\lim_{x\...
11
votes
2answers
361 views

Teaching limits and asymptotics at the same time

Having never been a mathematics educator, my question could be stupid and, if this is the case, please delete it. When I was young, from the very beginning of limits, we were teached that there are ...
20
votes
3answers
1k views

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

Let me explain. Suppose we want to calculate $\lim\limits_{n\to\infty} n^2-n$. Since this limit is indeterminate, one way to do it is to write it as $\lim\limits_{n\to\infty} n^2(1-1/n)$. Since $n^2$ ...
8
votes
6answers
313 views

Any metaphors/intuitions for a limit of a sequence?

I'm writing (together with a colleague) a minicourse on mathematical analysis (currently we want to cover the Weierstrass theorem on functions on compact intervals, so the aim is to present only the ...
13
votes
7answers
550 views

When should we get into limits in introductory calculus courses?

All of the calculus textbooks I've used (teaching at community colleges) start with the first chapter covering limits. (Perhaps after a review chapter.) I think this order is wrong. Historically, ...