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# Questions tagged [limits]

For question regarding the properties and evaluation of limits.

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13 votes
6 answers
5k views

### "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/...
• 2,039
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 ...
• 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 ...
• 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 ...
• 121
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 ...
• 425
-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 ...
• 1,144
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 ...
• 229
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 ...
• 223
1 vote
1 answer
331 views

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,...
• 828
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 ...
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 ...
-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 ...
-4 votes
7 answers
582 views

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 ...
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\...
• 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 ...
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 $n^2$ ...
• 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 ...
• 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, ...
• 21k