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Questions tagged [limits]

For question regarding the properties and evaluation of limits.

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13 votes
6 answers
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"Real life" examples of limits of functions at finite points

This is more specific than this similar question on math.SE, since I'm not satisfied with the answers there. Question: Can you provide an interesting, natural and simple example of some physical/...
Michael Bächtold's user avatar
0 votes
1 answer
175 views

What would constitute as a good justification of why a divergent limit is divergence for highschool teaching?

For example, consider $\lim_{x \to -\infty} \frac{x}{e^x}$, what would constitute as a good justification that the limit diverges too infinity? It's pretty easy to justify convergent limits ...
Babu's user avatar
  • 595
2 votes
2 answers
864 views

Process of finding limits for multivariable functions

I was tutoring a student today and they asked a question which made me curious. We were working on the following question together. After explaining that we must look at the limit along the x axis, I ...
Oofy2000's user avatar
  • 153
0 votes
2 answers
199 views

Example of a phenomenon from real life where there is a limit going to infinity

I haven't been able to find examples in real life where we have a function or sequence such that the limit goes to infinity when the independent variable goes to infinity. The only one so far is ...
Math Guy's user avatar
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3 votes
6 answers
1k views

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

tldr: There is a simple intuitive definition of a limit for monotone sequences, and I suggest that it can be used to motivate the (more complicated) standard definition. I am asking for feedback on my ...
Asaf Shachar's user avatar
-2 votes
1 answer
184 views

Finite sum of infinite series

I have two issues related to finite sum of infinite series, 1) How you would to describe 2 when you talk about the infinite geometric series 1+ 1/2 + 1/4 + 1/8 + ..... 2) How you would compare using ...
Janaka Rodrigo's user avatar
1 vote
1 answer
213 views

Introducing direct substitution in an intro calculus course

I'm revisiting the materials I've put together for students taking a non-proof-based intro to calculus, and my goal is for them to have a clear but rough sense of a limit as a bound (basically enough ...
Rax Adaam's user avatar
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7 votes
8 answers
5k views

Nice examples of limits to infinity in real life

I have to teach limits to infinity of real functions of one variable. I would like to start my course with a beautiful example, not simply a basic function like $1/x$. For instance, I thought of using ...
user700974's user avatar
1 vote
1 answer
331 views

Limit from both sides or from left? [closed]

Is it possible to write a problem statement as follows: A function $f$ is defined on $]0,1[$ as $f(x)=x$. Determine $\lim_{x\to 1}f(x)$. Or should one write always as: A function $f$ is defined on $]0,...
guest's user avatar
  • 21
6 votes
2 answers
893 views

Calculus limits taught in the US vs Spain?

So, I realize this can be a broad question, so I'll narrow it down. I have lived in Spain and own several Math textbooks from that country (the equivalent of 8th grade and high school Math). Has ...
Wasp's user avatar
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3 votes
3 answers
253 views

Differing Choices of $\delta$ in a Limit

In conceptually motivating the $\epsilon-\delta$ definition and proof of a limit, I realized a new way of choosing the $\delta$. For example, consider $\lim_{x\to 4}\sqrt{x}=2$. In the "standard ...
Aeryk's user avatar
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4 votes
1 answer
199 views

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 $f$?

When evaluating the limit of $f(x, y)$ as $(x, y)$ approaches $(x_0, y_0)$, we should or should not consider only those $(x, y)$ in the domain of $f(x, y)$ ? I am confused by different practices of ...
taiwanjizhan's user avatar
11 votes
3 answers
650 views

Terminology for parts of limit notation

When we talk about: $$\lim_{x\to{c}}f(x)=L.$$ Is there a formal name for the number "$c$"? I know that the notation means "$L$ is the limit of $f(x)$ as $x$ approaches $c$". It ...
Ari's user avatar
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11 votes
4 answers
532 views

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

L'Hôpital's rule for the indeterminate form $\frac00$ at finite points can be given a nice intuitive explanation in terms of local linear approximations. See for instance this textbook or this one. ...
Mike Shulman's user avatar
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4 votes
3 answers
191 views

(use of de L'Hospital's rule) How would you explain this limit to high school students?

Someone asks the following question: Determine the values for the real numbers $a$ and $b$ such that $$\lim_{x\to0}\frac{ae^x+b\cos x}{x}=2$$ The students directly apply de LHospital's rule for the ...
user12800's user avatar
14 votes
7 answers
2k views

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

$$\lim_{x\rightarrow a} f(x)=L$$ Which way should students best get in the habit of? The limit of $f(x)$, as $x$ approaches $a$, equals $L$ The limit of $f(x)$ equals $L$, as $x$ approaches $a$ The ...
Sat's user avatar
  • 331
4 votes
3 answers
371 views

Is there a pre-calculus introduction to the formal definition of a limit?

To give an example of what I mean, I'll answer a similarly worded question: “is there a pre-calculus introduction to the derivative?” I would say yes, since there already are the ideas of a slopes of ...
Sat's user avatar
  • 331
18 votes
5 answers
14k views

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

I would say it is not the Fundamental Theorem of Calculus, but rather some notion connecting limits and continuity, perhaps the $(\epsilon,\delta)$-definitions of limits and continuity. But I would be ...
Joseph O'Rourke's user avatar
9 votes
1 answer
851 views

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

Today in my calculus class we were going over L'Hopital's Rule and were dealing with limits of the following form $$h(x)=f(x)^{g(x)}$$ Three examples we considered are as follows: $(1)\; \...
Eleven-Eleven's user avatar
3 votes
1 answer
189 views

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

I've taught how to use $\epsilon, \delta$ to prove that a function is continuous at a point, and I'm about to teach how to prove that a function is uniformly continuous over an open interval. Usually,...
Alec's user avatar
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4 votes
3 answers
461 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 ...
pasaba por aqui's user avatar
4 votes
2 answers
283 views

How to explain the concepts of limits and continuity to non-mathematical students

How can I explain the fundamental concepts of limits and continuity to a student with a non-mathematical background? I am a PhD student in Mathematics working in Differential Geometry. As a part of my ...
user avatar
-4 votes
1 answer
248 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 ...
user37237's user avatar
-4 votes
7 answers
582 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}^\...
user37237's user avatar
9 votes
3 answers
2k 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 ...
benblumsmith's user avatar
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0 votes
1 answer
89 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 ...
Wilfred V's user avatar
8 votes
4 answers
396 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 ...
user avatar
17 votes
5 answers
414 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 ...
amarius8312's user avatar
8 votes
6 answers
2k 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 ...
Stephan Kulla's user avatar
4 votes
1 answer
396 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 $...
Ekaveera Gouribhatla's user avatar
33 votes
3 answers
2k 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 ...
user avatar
8 votes
5 answers
617 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\...
Jessica B's user avatar
  • 5,832
13 votes
2 answers
562 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 ...
Claude Leibovici's user avatar
21 votes
3 answers
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 $n^2$ ...
Javier's user avatar
  • 675
8 votes
6 answers
467 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 ...
mbork's user avatar
  • 1,299
13 votes
7 answers
2k 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, ...
Sue VanHattum's user avatar
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