Questions tagged [limits]
For question regarding the properties and evaluation of limits.
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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 ...
user507
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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 $n^2$ ...
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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 ...
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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 ...
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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 ...
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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/...
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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, ...
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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 ...
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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. ...
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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 ...
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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)\; \...
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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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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 ...
user2139
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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 ...
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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?
$$\lim_{x\...
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Any metaphors/intuitions for a limit of a sequence?
I'm writing (together with a colleague) a minicourse on mathematical analysis (currently we want to cover the Weierstrass theorem on functions on compact intervals, so the aim is to present only the ...
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Nice examples of limits to infinity in real life
I have to teach limits to infinity of real functions of one variable. I would like to start my course with a beautiful example, not simply a basic function like $1/x$. For instance, I thought of using ...
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Calculus limits taught in the US vs Spain?
So, I realize this can be a broad question, so I'll narrow it down. I have lived in Spain and own several Math textbooks from that country (the equivalent of 8th grade and high school Math). Has ...
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When evaluating the limit of $f(x, y)$ as $(x, y)$ approaches $(x_0, y_0)$, should we consider only those $(x, y)$ in the domain of $f$?
When evaluating the limit of $f(x, y)$ as $(x, y)$ approaches $(x_0, y_0)$, we should or should not consider only those $(x, y)$ in the domain of $f(x, y)$ ? I am confused by different practices of ...
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(use of de L'Hospital's rule) How would you explain this limit to high school students?
Someone asks the following question:
Determine the values for the real numbers $a$ and $b$ such that
$$\lim_{x\to0}\frac{ae^x+b\cos x}{x}=2$$
The students directly apply de LHospital's rule for the ...
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About the word "limit" used in calculus
In the introduction of the limits chapter of a scholar book I can read (translated and abstract) the following examples:
"the price of a product has a limit value that, from this price, the
number of ...
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How to explain the concepts of limits and continuity to non-mathematical students
How can I explain the fundamental concepts of limits and continuity to a student with a non-mathematical background?
I am a PhD student in Mathematics working in Differential Geometry.
As a part of my ...
user9721
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answer
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How can I explain $\lim_{x \to \infty} \frac{e^x+e^{-x}}{e^x-e^{-x}}$ using L'Hôpital's Rule?
I have given a problem in limits to my students:
$$\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 $...
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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 ...
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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 ...
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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 ...
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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,...
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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 ...
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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 ...
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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,...
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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 ...
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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 ...
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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 ...
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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 ...
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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}^\...
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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 ...