Limits Definition and 1000 Threads

I have a sequence of functions ##0\leq f_1\leq f_2\leq ... \leq f_n \leq ...##, each one defined in ##\mathbb{R}^n## with values in ##\mathbb{R}##. I have also that ##f_n\uparrow f##. Every ##f_i## is the limit (almost everywhere) of "step" functions, that is a linear combination of rectangles...

I know what the answers are, because this is all part of the notes from MIT OCW's 8.02 Electromagnetism course. In case you want to see the actual problem, it is example 2.3 that starts on page 18; the limits I am asking about are on page 20. How do I go about calculating the limits? Ie, what...

We have ##a_n## converges in norm to ##a## and a set ##S## such that for all ##n\ge 0## $$\sup_{s\in S} <a_n,s><+\infty .$$ Is it true that ##\sup_{s\in S} <a,s><+\infty##

Let ##\omega_1## be the first uncountable ordinal such that ##x## is an element of ##\omega_1## if and only if it is either a finite ordinal or there exists a bijection from ##x## onto ##\omega##. I want to define a matrix such that the matrix contains each element of ##\omega_1## only once. To...

In the homework I am asked to proof this, the hint says that I can use the triangle inequality. I was thinking that if both series go to a real number, a real number is just any number on the real number line, but how do I go from there,

Let ##L\in E##. By definition, there is a subsequence ##\{x_{n_k}\}_{k\in\mathbb{N}}## that converges to ##L##. There is a natural number ##N## s.t. if ##n_k\geq N##, ##L\in(x_{n_k}-1,x_{n_k}+1)\subset(\inf\{x_n\}-1,\sup\{x_n\}+1)##. Hence, ##E## is a bounded set. If ##E## is a finite set, then...

This was the question, The above solution is the one that I got originally by the question setters, Below are my attempts (I don't know why is the size of image automatically reduced but hope that its clear enough to understand), As you can see that both these methods give different answers...

See attachment for question and math work.

Use the properties of limits to find the limit. lim [x(x − 1)(x + 10)] x→−1 lim [x(x^2 + 9x - 10)] x→−1 lim [x^3 + 9x^2 - 10x] x→−1 lim (x^3) + lim (9x^2) - (10x) x→−1 x→−1 x→−1. (-1)^3 + 9(-1)^2 - 10(-1) -1 + 9 -(-10) -1 + 9 + 10 -1 + 19 = 18 The limit is 18. You...

Use properties of limits to find the limit. lim (-3x + 1)^2 x→0 [lim (-3x + 1) as x→0 ]^2 [-3•lim(x) as x→0 + lim (1) as x→0]^2 [-3•0 + 1]^2 [0 + 1]^2 [1]^2 = 1 The limit is 1.

:: Tables and Limits Complete a table for f(x) = x + 3 as x→2 from the right and left. As x tends to 2 from the left side, the given values for x are: 1.9, 1.99, 1.999. As x tends to 2 from the right side, the given values for x are: 2.001, 2.01, 2.1. Let me see if I can do this. I think...

distance (metres) Angle of elevation (degree) 950 7 1500 12 1800 15 3300 20 I think I don't need to use information about 900 metres because the path starts from elevation of 1000 metres. I imagine the distance will be the hypotenuse of a triangle and the height of a certain location...

Hello everyone. How are you? I want to learn calculus so badly. I plan to do a self-study through calculus l, ll, and lll. Before I think so far ahead, I need a clear, basic definition of the concept of a limit. Textbook language is never easy to grasp unless the student is gifted. I am not...

We sometimes write that \sin x=x+O(x^3) that is correct if \lim_{x \to 0}\frac{\sin x-x}{x^3} is bounded. However is it fine that to write \sin x=x+O(x^2)?

Problem: If sequence ## (a_n) ## has ##10-10## as partial limits and in addition ##\forall n \in \mathbb{N}.|a_{n+1} − a_{n} |≤ \frac{1}{n} ##, then 0 is a partial limit of ## (a_n) ##. Proof : Suppose that ## 0 ## isn't a partial limit of ## (a_n) ##. Then there exists ## \epsilon_0 > 0 ## and...

How would I determine the following limit without substitution of large values of x to see what value is approached by the complex function? ## \lim_{x \rightarrow +\infty} {\dfrac {2^{x}} {x^{2} } } ## where ## x\in \mathbb{R}##

First I quote the text, and then the attempts to solve the doubts: "Proof of the Chain Rule Be ##f## a differentiable function at the point ##u=g(x)##, with ##g## a differentiable function at ##x##. Be the function ##E(k)## described this way: $$E(0)=0$$...

Given equation and conditions: ##\boldsymbol{x^2+2(k-3)x+9=0}##, with roots ##\boldsymbol{(x_1,x_2)}##. These roots satisfy the condition ##\boldsymbol{-6<x_1,x_2<1}##. Question : ##\text{What are the allowable values for}\; \boldsymbol{k}?## (0) Let me take care of the determinant first...

This thread contains posts that were off-topic in another thread No wonder you have trouble with printers...

The question about how using limits can give us the exact slope of a line tangent to a curve is something so far I haven't quite been able to grasp. I do intuitively understand how using limits can get us so close to the exact slope that any difference shouldn't matter in the real world because...

Problem: The sphere is parametrized in cylindrical coordinates by: x = r cosθ y = r sinθ z = (1-r^2)^1/2 and intersected by the cone (x-1)^2 +y^2 = z^2. find the area of the sphere enclosed by the cone using the equation: da = r/(1-r^2) dr dθ Attempt at solution: from the equations for the...

Up and Atom gives a look at some Calculus tools which a layman can follow.

Hey, so I have the following problem: I'm trying to prove that the limit doesn't exist (although I'm not sure if it does or not) so: along y=mx -> x=y/m: , which is 0 for all k≠0. along y^n it's the same and I'm not sure what I should do next. Could I set x = sin(y)? If I can, then the limit...

As I can't find it in my basic calculus book, can someone please tell me what book that taught a limit theory and problem like these? Thank you

By ZFC, the minimal set satisfying the requirements of the axiom of infinity, is the intersection of all inductive sets. In case that the axiom of infinity is expressed as ∃I (Ø ∈ I ∧ ∀x (x ∈ I ⇒ x ⋃ {x} ∈ I)) the intersection of all inductive sets (let's call it K) is defined as set K = {x...

Hi. There has been a fair amount of research into electric generators and motors with superconductive coils. If traditional iron cores is used that obviously limits the power density because of the iron cores magnetic saturation point. But for coreless/ironless designs i don't understand what...

Hello there.Is there any function or sequence that has no limits at any point? I am not necessarily talking about functions on euclidean spaces, they could be on topological spaces in general.Also, we have homeomorphism that is about I think mostly continuity, diffeomorphism about...

Hi, I apologise as I know I have made similar posts to this in the past and I thought I finally understood it. However, this solution seems to disagree on a technicality. I know the answer ends up as 0, but I still want to understand this from a conceptual point. Question: Evaluate the line...

Hey there, I'm aware this is a bit of a stupid question, and I think that I understand the principle fundamentally, however, my intuition is still having a little trouble catching up, and I'm trying to figure out if it is because of an important detail that I have missed/misinterpreted. I think...

Milky Way centre cannot be seen because of Great Rift, but it is known as a loud radio source Sagittarius A since 1930s. What would be the absolute magnitude of Sagittarius A if it could be seen? Visual magnitude? Not all galaxies have loud centres. Large Magellanic Cloud does not show a sign...

An example (I think) of creating a phenomena that appears to propagate faster than the speed of light would be to have a line of people holding flashlights and giving each person a schedule of when to blink his light. With proper schedule we could create the illusion that point of light is...

I was thinking - and reading a bit - about the size limit on accelerated frames, and there is an interesting and relevant result I found. If we rephrase the question from "is there a size limit on an accelerated frame" to "is there a size limit on an accelerated body in irrottational born...

Once again I'm looking at a project that is challenging my engineering limits! I have a type of metering device that has an electromagnetic coil that actuates it, it's designed to be relatively linear motion depending on amperage applied through it, and it has a variable resistor as a sensor for...

$\displaystyle\lim_{x \to 0}\dfrac{1-\cos^2(2x)}{(2x)^2}=$ by quick observation it is seen that this will go to $\dfrac{0}{0)}$ so L'H rule becomes the tool to use but first steps were illusive the calculator returned 1 for the Limit

by observation I choose (c) since the limit values may not be =

Hello. I bought "Calculus Made Easy" by Thompson and it got me thinking about something I wondered about before. This question is a bit hard for me to articulate, but I'll do my best: When we are trying to find the limit as change in x approaches zero of dy/dx, we take smaller and smaller...

What do members speculate, using restrained imagination of what may be possible, may be the highest level of knowledge and ability possible for an ancient race of artificial intelligence entities millions of years evolved that have long mastered whatever organic brain consciousness is and gone...

From the equations, I can find Jacobians: $$J = \frac {1}{4(x^2 + y^2)} $$ But, I confuse with the limit of integration. How can I change it to u,v variables? Thanks...

Because the limit of the integral is multi-variable, which is not explained at the ML Boas's example, I tried to start from the basic. First, I use: $$\frac {dF}{dx}=f(x) \Rightarrow \int_a^b f(t) dt = F(b) - F(a)$$. In my case now: $$\int_{u(x)}^{v(x,y)} f(t) dt = F(v(x,y)) - F(u(x))$$ So...

I have a question like this; I selected lambda as 4 (I actually don't know what it must be) and try to make clear to myself like these limits (1,2) and (2,4) is x and y locations I think :) If I find an answer for part one of the integral following, I would apply this on another: My...

I am not sure about finding the limit of the integral when it comes to finding the CDF using the distribution function technique. I know that support of y is 0 ≤y<4, and it is not a one-to-one transformation. Now, I am confused with part b), finding the limits when calculating the cdf of Y...

Is this anyhow possible ? The system would be a wave equation modelized by a finite elements basis in space and time. Is there any method to do the limit discretization->continuum with paper and pencil ?

if someone can concur that'd be great; also, is there any way for me to check myself in the future?

Let: \begin{align} r&=\sqrt{a^2 + p^2 - 2ap \cos \theta}\\ s&=a\\ t&=p\\ f(r) &= \text{continuous function of } r\\ g(s) &= \text{continuous function of } s\\ \end{align} Consider the expression: \begin{align} \int_{q'}^q \int_{b'}^b g(s)\ \int_{s-t}^{s+t} f(r)\ dr\ ds\ dt\ \end{align} We...

Hey! :o Could you give me a hint how to prove the following statements? (Wondering) Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be differentiable (or twice differentiable). $\left.\begin{matrix} \displaystyle{\lim_{x\rightarrow +\infty}f(x)=\ell} \ (\text{or } \displaystyle{\lim_{x\rightarrow...

(NOTE: I have had a few similar postings lately on this subject, but they were much broader in scope, so I am posting only for this particular case; everything else has been figured out.) If given that limx -> a f( x ) = +∞ limx -> a g( x ) = +∞ what is the epsilon-delta formulation for...

I was looking at some websites that show the proof of addition of limits for a finite output value, but I don't see one for the case of infinite output value, which has a different condition that needs to be met - i.e., | f( x ) | > M instead of | f( x ) - L | < ε...

AIUI, this is a law of proofs: lim x→a f( g( x ) ) = f( lim x→a g( x ) ) I have searched for an explanation of this proof, but have been unable to find one, although I did find a page that was for certain types of functions of f( x ), just not a proof for a function in general.