What is Limits: Definition and 1000 Discussions

Homework Statement Hello, thank you in advance for all help. This is a limit problem that is giving me a particularly hard time. Homework Equations For what values of a and b is f(x) continuous at every x? In other words, how to unify the three parts of a piecewise function so that there are...

Let ## f(x)=\begin{cases}1 &if \ x=\frac{1}{n} \ where \ n\epsilon \mathbb{Z}^{+}\\ 0 & \mbox{otherwise}\end{cases}## (i) Show that ##c\neq 0## then ##\lim_{x \to c}f(x)=0## (ii) Show that ##\lim_{x \to 0}f(x)## does not exist. I attempted to answer the question: I think we have to show...

Homework Statement Let an → 2. Prove from first principles (i.e. give a direct ε-N proof) that an2 → 4. Homework EquationsThe Attempt at a Solution I have tried considering |an-2|2 and considering that |an2-4| = |(an+2)(an-2)| but I could not get either of these methods to work. Would someone...

lim(9-x) as x->4 = 5 I thought I was supposed to do this: 9-4=5 5=5 But apparently I was supposed to use delta and epsilon? I'm not sure how to find either of these. I know you find epsilon first but I'm really confused so if anyone knows just HOW to find it, that would be extremely helpful...

Given a function f(x), a point x0, and a positive number E (epsilon), write the limit then find delta>0 such that for all x 0< |x-x0| < delta -> |f(x)-L| < E f(x) = 3-2x, x0=3, E=.02 Here is my attempt: Lim (3-2x) as x->3 = -3 -.02 < |3-2x - 3| <.02 -.02 < |-2x| < .02 .01 > x > -.01 -2.99 > x-3...

The website rules say that when people help they are not allowed to include answers? But how am I supposed to check my answers... anyone else have this problem?

Hello

Would it be a legitimate (valid) proof to use an \epsilon-\delta limit approach to prove the fundamental theorem of calculus? i.e. as the FTC states that if f is a continuous function on [a,b], then we can define a function F: [a,b]\rightarrow\mathbb{R} such that F(x)=\int_{a}^{x}f(t)dt Then F...

I know that steel and titanium have fatigue limits. Just to clarify, metals or alloys with fatigue limits are metals that - as long as they experience pressures that lower than the limits - can last "indefinitely". Aluminum, for example, does NOT have a fatigue limit. No matter how small the...

Homework Statement Find the limit analytically: lim (x->4) [(x-3)^0.5 -1]/[x-4] Homework Equations The Attempt at a Solution lim (x->4) [(x-3)^0.5 -1]/[x-4] = lim (x->4) ([(x-3)^0.5 -1]/[x-4]) * [((x-3)^0.5 +1)/((x-3)^0.5 +1)] = lim (x->4) [x - 3 -1]/[(x-4)(x-3)^0.5 +1)] = [4 - 3...

I read in a calculus book that. "Given ##\lim_{x \to a}\frac{f(x)}{g(x)} = c(c\neq 0)##, when ##\lim_{x \to a}g(x) = 0##, then ##\lim_{x \to a}f(x) = 0##. Why is this true?

Homework Statement Lim (x,y,)--> (1, 3) of (x^2-1)/(xy-y) Homework Equations I know that the answer is 2/3 according to wolfram alpha multivariable limit calculator. The Attempt at a Solution So this is my first time doing multivariable limits, I've studied the following: 1) Direct...

I am curious about the process of evaluating a limit. Firstly, I know that if a function ##f(x)## is continuous then one can usually just plug in the the number that ##x## is approaching in the limit, since criteria for a continuous function is that ##\lim_{x \to a}f(x) = f(a)##. However, what...

Say we have \int_{0}^{\omega _{D}} \frac{\hbar \omega^{3}}{exp (\frac{\hbar\omega}{kT}) - 1} d\omega let x = \frac{\hbar\omega}{kT} if we sub in we get (\frac{kT}{\hbar})^{3} (\frac{kT}{\hbar}) \int_{0}^{\omega _{D}} \frac{x^{3}}{exp (x) - 1} dx my question is how would the limits and...

I am (slowly) learning thermodynamics. I find a lot of it puzzling and makes me formulate many conjetures; I hope any of you can help me with this one about heat exchange. Let's start with this system: There is a closed hose loop filled with water; its temperature gradient goes from cold to...

Homework Statement Find lim_{x->- \infty} \; \frac{(x^6+8)^{1/3}}{4x^2+(3x^4+1)^{1/2}} Homework Equations N/A The Attempt at a Solution Factoring out \frac {(-x^6)^{1/3}}{-x^2} leaves me with \frac{(-1-8x^{-6})}{-4+(3+x^{-4})^{1/2 }} Taking the limit at infinity gives me...

I need some help in understanding the reasoning and analysis in the solution to Exercise 4.5 in Robert C. Wrede and Murray Spiegel's (W&S) book: "Advanced Calculus" (Schaum's Outlines Series). Exercise 4.5 in W&S reads as follows: https://www.physicsforums.com/attachments/3918 The...

f(x) = x^3 - 12x^2 + 44x - 46 x from 1 to 7 The attempt at a solution: f(1) = -13 f(2) = 2 f(4) = 2 f(5) = -1 f(6) = 2 So naturally, the answer should be: (1,2) U (4,5) U (5,6) right? Well, it didn't accept this answer. I think there is something wrong with whatever that is accepting the...

∞Homework Statement If Lim f(x) and Lim g(x) both exist and are equal x→a x→a then Lim[f(x)g(x)]=1 x→a Homework Equations No relevant equations are required in this problem. To determine whether the statement is true or false [/B]The...

I have just posted an edit to my (very) recent post: http://mathhelpboards.com/analysis-50/apostol-continuity-amp-differentiabilty-14190.htmlin the Analysis Forum. I am having trouble with the following Latex expression:\text{lim}_{x \rightarrow c} f^* (x) = \text{lim}_{x \rightarrow c}...

Homework Statement I need help finding the limit of the differential equation. (dx/dt) = k(a-x)(b-x) that satisfies x(0)=0 assuming a) 0<a<b and find the limit as t->infinity of X(t) b) 0<a=b and find the limit as t->infinity of X(t) Homework Equations none The Attempt at a Solution I...

I don't understand the relationships between the integration limits of Maxwell Equations (specifically the ones in integral form in matter) Is this related to Stokes/Gauss' Theorems? or something else?

I'm using the method of characteristics to solve a pde of the from ## au_{x}+bu_{y}=c## where ## a=\frac{dx}{d \tau} , b= a=\frac{dy}{d \tau}, c=a=\frac{du}{d \tau}## where initial data is parameterised by ##s## and initial curve given by ##x( \tau)=x_{0}(s)##, ##y( \tau)=y_{0}(s)## and ##u(...

If there were an infinite number of universes will it be true to say that every conceivable (imagination) life form exists somewhere? Example, somewhere in a universe a 3 headed dragon exists - or does mathematical limit apply in this case?

Homework Statement Dear Mentors and PF helpers, Here's my question, I see these on my textbook but couldn't really understand how they derived this short cut. Please show me how they got to these. Thank you for your time. Homework Equations These is what I understand from now. The...

I know I'm not smart enough to make it as a physicist. And this isn't going to be another, "Convince me to be a physicist" thread. I just want to know how others approach the problem of simply not being good enough to do something you want to do. I had dreams the last couple years of getting a...

Hi, I'm looking at deriving the limits of ##\dot{a}## as ## a-> \infty ## , using the Friedmann equation and conservtion of the ##_{00}## component of the energy momentum tensor for a perfect fluid. Both of these equations respectively are: ## \dot{a^{2}}=\frac{8\pi G}{3} \rho a^{2} + | k |##...

Homework Statement Determine the following limit in terms of the two real-valued parameters A and B: lim_{x \rightarrow 0} (\frac{Ae^{A/{x^2}}+Be^{B/{x^2}}}{e^{A/{x^2}}+e^{B/{x^2}}}) Homework Equations L'Hopital's rule The Attempt at a Solution I first divided by e^{A/{x^2}} in both...

I was going through some important points give in my textbook and I saw this: ##\log_e x < \sqrt x## How did they get this? I know calculus so you can show this using differentiation, etc. One possible way is that they took ##f(x)=\sqrt x-\log_e x## And tried to prove it is always greater than zero.

Homework Statement Using sandwich theorem evaluvate: $$\lim_{x\rightarrow \infty} \frac{x+7sinx}{-2x+13}$$ Homework Equations Sandwich theorem The Attempt at a Solution ##-7 \leqslant 7sinx \leqslant 7## ##x-7 \leqslant x+7sinx \leqslant x+7## Now my doubt: I want to divide the expression by...

As far as i know, measurement devices present measurements based on something that affects the device's particles, for instance, forces, heat, tension, voltage... My question is, given that every change of position of any particle may affect the particles of the measurement device, why can't we...

Homework Statement In writing the definition of ##e## i.e. ##e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n##, why do we denote the variable by 'n' despite the fact that the formula holds for n∈(-∞,∞)? Is there any specific reason behind this notation i.e. does the variable have...

(I have posted this in this section, rather than homework, because I hope to improve my general understanding of methods of finding limits through these problems.) 1: \lim_{x\rightarrow0} (\frac{cosec(x)}{x^3} - \frac{sinh(x)}{x^5}) I don't really know what to do with this one. I tried...

Homework Statement [/B] I have to find the radius of convergence and convergence interval. So for what x's the series converge. The answer is supposed to be for every real number. So the interval is: (-∞, ∞). So that must mean that the limit L = 0. So the radius of convergence [ which is...

Am I correct in assuming that if there is a potential present and it is not infinite then integrals will always be made from minus infinity to infinity, but where an infinite potential exists then the integral will depend on the size of the confinement area? Sorry to be a little disambiguous...

Homework Statement sketch the solid region contained within the sphere, x^2+y^2+z^2=16, and outside the cone, z=4-(x^2+y^2)^0.5. b) clearly identifying the limits of integration, (using spherical coordinates) set up the iterated triple integral which would give the volume bounded by the...

Hello, I am working towards an extremely difficult real analysis problem. The statement is as follows: Prove that if $\lim_{x \to a} f(x) = l$ and $\lim_{x \to a} g(x) = m$ then $\lim_{x \to a} \max(f(x), g(x)) = \max(l, m)$ Some definitions: $$\max(f, g)(x) = \frac{f + g + |g - f|}{2}$$...

Homework Statement [/B] lim x -> 0 2. Homework Equations Taylor series for sin cos e and ln () The Attempt at a Solution I tried expanding the sine to 3-degree, and everything else 2-degree. I ended up with this: Now the problem is that WolframAlpha says it should be -6/25. Now if...

Hello all I have a general question. When I look for a limit of a rational function, there is this rule of dividing each term by the highest power. I wanted to ask if I should divide by the highest power, or the highest power in the denominator, and why ? I have seen different answers in...

Homework Statement Evaluate the limit 1 1 1 lim ∫ ∫ ... ∫ cos^2((pi/2n)(x1 + x2 +... xn))dx1 dx2 ... dxn 0 0 0 n→∞Homework Equations Well, I know that we can change this using a double angle rule, so that the integrals become 1/2 + 1/2 cos (2*pi/2n)(x1 +...

I'm still very new to the world of calculus and physics, and while I understand how to do limits and such, I don't understand the usefulness of them, and I'm hoping some of you can help me understand why limits are important and why we bother with them. Here's my understanding of them...

Homework Statement I am studying for a calculus test tomorrow on this website (http://archives.math.utk.edu/visual.calculus/6/index.html). I am working on the limit comparison test problems but I am unfamiliar with the form they use in their solutions. For example: Limit comparison test (prove...

Say we do physics in a very large box of side L. Using the proper superposition of a countable number of momentum eigen states can we write down the wave function of a localized high energy particle in a box? If so, assume the number of superposed momentum states is N. Now randomly throw away...

Mod note: Fixed the LaTeX. The closing itex tag should be /itex, not \itex (in brackets). I find it easier to use # # in place of itex, or $ $ in place of tex (without the extra space). Homework Statement Prove \lim_{x \to 0} \frac{x}{\sin^2(x) + 1} = 0 Homework Equations Given below: The...

Homework Statement Suppose that limit x-> a f(x)= infinity and limit x-> a g(x) = c, where c is a real number. Prove each statement. (a) lim x-> a [f(x) + g(x)] = infinity (b) lim x-> a [f(x)g(x)] = infinity if c > 0 (c) lim x-> a [f(x)g(x)] = negative infinity if c < 0 Homework Equations...

In the integral integral(1,infinity) e^(-sqrt(x)) / sqrt(x) STEP 1: I let u = -sqrt(x) du = -1/(2sqrt(x)) then my lower bound u = -1 then my upper bound u = -infinity -2 integral(-1,infinity) e^u du I would then switch the order of the integration bounds and multiply by -1My question is...

Homework Statement Show that ##\lim_{x \to a} f(x) = L## if and only if ##\lim_{x \to 0} f(x+a) = L## Homework Equations - The Attempt at a Solution For the forward direction (ie ##1 \Rightarrow 2##), I tried to first assume that 1. holds true (ie ##\forall \epsilon>0, \exists \delta>0...

Homework Statement The problem is: f(x) = {ax^2 - b / (x - 2), x < 2 1/4(x^2) - 3c, x >= 2 If f is differentiable at x=2, what are the values of a, b, and c? In case it is hard to see, I have provided an image 2. Homework Equations I know that the limit definition of a...

Hello (: Can someone help me calcultating this limit (as x → ∞) This is the function: If I just kept deriving sin(e^x) nothing really happened. So I tried using L'Hopital but partly, i guess: so: (1 or -1) × 1 + 0 + 0 / 0 + ∞ + ∞The answer must be 1 (according to wolfram alpha) but i...

Homework Statement Find the limit of the following sequence: Homework Equations \lim_{n \rightarrow + \infty} \sqrt[4] {2n + 1} - \sqrt[4] {n + 1} The Attempt at a Solution I've tried multiplying the first radical by ## \frac{ \sqrt[4] {2n - 1} } { \sqrt[4] {2n - 1} } ## to make the radical...