Limit Definition and 999 Threads

Homework Statement given a geometric sequence sin(x),sin(2x), . . . c) find for which values of x∈(0,π) this sequence converges and calculate its limit Homework Equations |q|<1 or -1<q<1The Attempt at a Solution Ok so in part a) and b) i calculated the quotient and found out that...

$$\lim_{{x}\to{\infty}}\frac{\sqrt{16{x}^{4}+64 {x}^{2} }+x^2}{2x^{2} - 4}=\frac{5}{2}$$ I tried to solve this by dividing all terms by$x^4$ but then the denomator will go zero.

This was just very basic, I have accepted it in just a heartbeat, but when I tried to chopped it and examined one by one, somethings fishy is happening, this just involved \int_{0}^{\infty}x e^{-x}dx=1. Well, when we do Integration by parts we will have let u = x du = dx dv = e^{-x}dx v =...

Homework Statement Hi, the problem is imply to show the following \lim_{n\rightarrow \infty} 10^n e^{-t} \sinh{10^{-n}t} = \lim_{n\rightarrow \infty} 10^n e^{-t} \sin{10^{-n}t} = te^{-t} How can I do this? Just a hint or a first step would be great, thanks :) Homework EquationsThe Attempt...

Hi everybody, I have this function to study ##\frac{(x+1)}{arctan(x+1)}## I need the limit to infinity,it's oblique and I have to find q,from y=mx+q. so q=lim(x->inf) ##\frac{(x+1)}{arctan(x+1)} -2x/\pi## I don't know how to solve it.the limit gives infinity to me.but calculators online give...

How to find the ## \lim_{x \to \infty} (\sqrt{ax^2 + bx + c} - \sqrt{ax^2 + px + q}) = \frac{b - p}{2 \sqrt{a}} ## ? Here is what I get ## \lim_{x \to \infty} (\sqrt{ax^2 + bx + c} - \sqrt{ax^2 + px + q}) ## ## = \sqrt{\lim_{x \to \infty} ax^2 + \lim_{x \to \infty} bx + \lim_{x \to \infty}...

So if the universe expansion is accelerating due to dark energy, does that mean that (assuming there is) one end of the universe relative to the other end of the universe will see it moving away at speeds greater than the speed of light? Or is the expansion capped by relativity? Or does the...

Spivak proves that limit of function f (x) as x approaches a is always unique. ie...If lim f (x) =l x-> a and lim f (x) =m x-> a Then l=m. This definition means that limit of function can't approach two different values. He takes definition of both the limits. He...

Homework Statement Homework EquationsThe Attempt at a Solution I tried using the rule of multiplying with the "conjugate", for example what's above multiplied by (√n^3+3n)+(√n^3+2n^2+3)/(√n^3+3n)+(√n^3+2n^2+3). But I'm left with a huge mess :( I also tried dividing the top and the bottom by...

I have a question for determining the limit of a function with two variables. My textbook says that the limit (x,y)->(0,0) of 4xy^2/(x^2+y^2)=0. This is true if we evaluate the limit if it approaches along the x-axis (y=0) or the y-axis (x=0) or any line on the plane y=kx. I am wondering if...

$\displaystyle \lim_{x \to 0} \frac{x^2-\sin^2{x}}{\tan(3x^4)}$ How do you calculate this one? L'hopital gives me $\displaystyle \lim_{x \to 0} \frac{2x\cos^2(3x^4)-\sin{2x}\cos^2(3x^4)}{12x^3}$

Homework Statement Consider the sequence given by b_{n} = n - \sqrt{n^{2} + 2n}. Taking (1/n) \rightarrow 0 as given, and using both the Algebraic Limit Theorem and the result in Exercise 2.3.1 (That if (x_n) \rightarrow 0 show that (\sqrt{x_n}) \rightarrow 0), show \lim b_{n} exists and find...

With this one I don't know where to start $\displaystyle \lim_{h \to 0} \frac{5^{\sin{h}}-1}{\tan{h}}$

How do you calculate the limit $\displaystyle \lim_{x \to \infty}\frac{x}{\cosh{x}}$

Dividing by the highest power for $\displaystyle \lim_{x \to 5^{-}}\frac{x^{100}-4x^{99}}{x-5}$ I get $\displaystyle \lim_{x \to 5^{-}}\frac{x^{100}-4x^{99}}{x-5}= \lim_{x \to 5^{-}}\frac{1-4/x}{1/x^{99}-5/x^{100}}$ However the denominator goes to $0$ whereas the numerator goes to $1-4/5$ Why...

How do you show that $\displaystyle \lim_{x \to \infty} \frac{50x^{10}+100}{x^{11}+x^6+1}=0$ What I tried: $\displaystyle \lim_{x \to \infty} \frac{50x^{10}+100}{x^{11}+x^6+1} =\lim_{x \to \infty} \frac{50+100/x^{11}}{1+1/x^{5}+1/x^{11}} = \frac{50+0}{1+0+0} = 50.$ But this is wrong. (Angry)

I'm trying to find $\displaystyle \lim_{x \to 20^{+}}\frac{5x^3+1}{20x^3-8000x}$ $\displaystyle \lim_{x \to 20^{+}}\frac{5x^3+1}{20x^3-8000x} =\lim_{x \to 20^{+}}\frac{5+1/x^3}{20-8000/x^2} = \frac{5+\lim_{x \to 20^{+}}1/x^3}{20-\lim_{x \to 20^{+}}8000/x^2} =...

Homework Statement Calculate the limit $$lim_{s,t→∞} R_X(s, s+t) = lim_{s,t→∞}E(X(s)X(s+t))$$ for a continuous time Markov chain $$(X(t) ; t ≥ 0)$$ with state space S and generator G given by $$S = (0, 1)$$ $$ G= \begin{pmatrix} -\alpha & \alpha \\ \beta & -\beta\...

Hi, my book says that $\lim_{{n}\to{\infty}} {n}^{p}U_n \rightarrow A \lt \infty, p \gt 1 $ means that $U_n \lt \frac{A}{{n}^{p}} $, which I can see But apparently $ \lim_{{n}\to{\infty}}n U_n = A \gt 0 $ means that $ U_n \gt \frac{A}{n} $ I know this is going to sound like a stupid...

Joos asserts on page 31 https://books.google.com/books?id=btrCAgAAQBAJ&lpg=PP1&pg=PA31#v=onepage&q&f=false that $$\nabla \times \mathfrak{v} = \lim_{\Delta \tau \to 0} \frac{1}{\Delta \tau }\oint d\mathfrak{S}\times \mathfrak{v}$$ I tried to demonstrate this, and neglected to place the surface...

Homework Statement use cylindrical coordinates to find the volume of the solid which is under z=xy, above xy-plane and inside the cylinder x^2+y^2=2x Homework EquationsThe Attempt at a Solution \int_{0}^{pi/2} \int_{0}^{2cos\theta} \int_{0}^{r^2\cos\theta\sin\theta} r\, dz \, dr \, d\theta...

For a state |\Psi(t)\rangle = \sum_{k}c_k e^{-iE_kt/\hbar}|E_k\rangle , the density matrix elements in the energy basis are \rho_{ab}(t) = c_a c^*_be^{-it(E_a -E_b)/\hbar} How is it that in the long time limit, this reduces to \rho_{ab}(t) \approx |c_a|^2 \delta_{ab} ? Is there some...

Homework Statement If we have a number sequence such that: a0, a1 are given, and every other element is given as ##a_n=\frac{(a_{n-1} + a_{n-2})}{2} then express an in terms of a0, a1 and n , and fin the limit of an Homework EquationsThe Attempt at a Solution If i try to express a3 in terms of...

I realized that (i + i/n)n approaches 4 discrete values (e, ei, -e and -ei) as n approaches infinity (if n is integer). (If I take that i2 = 1, then it approaches two discrete values (e and ei)). Does this kind of "multivalued limit" have some other name so I can learn more about it or where it...

Homework Statement Consider the dynamical system: $$\dot{r}=-ar^4+ar^3+r^6-r^5+r^2-r~;~~\dot{\theta}=1$$ Find all fixed points and limit cycles for: a) ##~~a=2## b)##~~a<2## c)##~~2<a<2\sqrt{2}## Homework Equations Not applicable. The Attempt at a Solution For all three values/ranges...

Besides solar wind, what is the limit of a natural wind on planets? I know it's based on rotation and atmospheric pressure differences, but is there an upper limit to how fast wind can get on a planetary scale? I would imagine that the speed of sound would justify a limit, but that doesn't...

So, according to answer sheet, the answer is 1... The question is : limit as x approaches infinity of : squareroot( x^2 + x ) - squareroot( x^2 - x) I tried to put it in a limit calculator, but the steps shown are very complex and don't even involve l'hopital's rule... I think the solution...

How much current can be forced through a photovoltaic (solar) cell before its performance starts to deteriorate? In certain accelerated lifetime testing 1.25xIsc is forced through a solar module while it sits inside a chamber (no incident light). Is there a theoretical limit to the amount of...

I am currently studying optical microscope and discover that the axial resolution is limited as r(z) = 2pi / (NA)^2. However, while I got hints that it is due to the Rayleigh's limit, I can't derivative the equation using numerical method. It would be huge thanks if anyone can help me on the...

Homework Statement I have lim of n > infinity (1+1/n)^n Homework EquationsThe Attempt at a Solution I know that I must use l'hospital rule and setting ln y = n ln (1+1/n) And after lim n ln (1+1/n) as n approaches infinity. After what do I do ?

The classical problem of radiation reaction classical electrodynamics seems to be a huge mess with no good answer. There is no even consensus of the very form of the Newton law "predicted" theory. So, the question of this thread is: Does the classical limit of QED say something about this...

Homework Statement An engineer is measuring a quantity q. It is assumed that there is a random error in each measurement, so the engineer will take n measurements and reports the average of the measurements as the estimated value of q. Specifically, if Yi is the value that is obtained in the...

Let $\lim_{{k}\to{\infty}}d\left({x}_{m\left(k\right)},{x}_{m\left(k\right)-1}\right)=\varepsilon$ and $\lim_{{k}\to{\infty}}d\left({x}_{n\left(k\right)},{x}_{m\left(k\right)}\right)=\varepsilon$...Can we say that...

Hello, If I have a momenta pμ=(E,px,py,pz) and transform it via lorentz boost in x-direction with velocity v I'll get for the new 0th component E′=γE+γvpx why is this in the limit of low velocities the same as transforming the energy by a galilei transformation with velocity v? For γvpx i get...

Let ##X_i## are i.i.d. and take -1 and +1 with probability 1/2 each. How to prove ##\lim_{n\rightarrow\infty}{\sum_{i=1}^{n}{X_i} }##does not exsits (even infinite limit) almost surely. My work: I use cauchy sequence to prove it does not converge to a real number. But I do not how to prove it...

so a quick Q. the equation for charging a capacitor seems to indicate that charge (watts) will always be charging the capacitor, but is it true that as t⇒∞ the charging actually stops and the state of equilibrium is quantized?

I am struggling to properly understand the \varepsilon-\delta definition of limits. So, f(x) gets closer to L as x approaches a. That is okay. However, taking the leap from there to the \varepsilon-\delta definition is something I have never really been able to do. Why is the formulation we...

Hi guys, I attempted to prove this theorem, but just wanted to see if it a valid proof. Thanks! 1. Homework Statement Prove that x is an accumulation point of a set S iff there exists a sequence ( s n ) of points in S \ {x} that converges to x Homework Equations N * ( x; ε ) is the x -...

Homework Statement Ground state energy is set at 0. E_n=\left(1-\frac{1}{n+1}\right)\in with no degeneracy (\Omega(n)=1); (n=0,1,2...) Write down the partition function and look for its limit when kt \gg \in\\ kt \ll \in Homework EquationsThe Attempt at a Solution Partition function for this...

Hello! I was wondering if anyone could expand upon and help me with this as I'm struggling "Use continuity to evalute $$\lim_{{x}\to{\pi}}\cos(x+\sin(x))$$" I do remember faintly how to do limits of "normal" numbers, but with trig I did not learn at all so I'm confused. This is same as finding...

Hello everyone, I am currently considering a set of random variables, \vec{x} = [x_1,x_2,...x_N] which are know to follow a multivariate normal distribution, P(\vec{x}) \propto \mathrm{exp}(-\frac{1}{2}(\vec{x}-\vec{\mu})^\mathrm{T}\Sigma^{-1}(\vec{x}-\vec{\mu})) The covariance matrix Σ and...

Homework Statement Let ##f## be defined by ##f(x) = x## if ##x## is rational, and ##f(x) = -x## if ##x## is irrational, prove that ##\lim_{x\to a} f(x)## exists if and only if ##a=0##. Homework Equations Epsilon-delta definition of a limit: ##\lim_{x\to a}f(x) = l## means that for every...

Homework Statement Homework Equations lim(x,y)->(a,b)f(x,y) continuous at (a,b) if lim(x,y)->(a,b)f(x,y)=f(a,b) Squeeze theorem if lim a=lim c and lim a<= lim b <= lim c then lim b= lim c The Attempt at a Solution I proved that all the limits exist but somewhat the functions aren't all...

I have a question regarding GRE general as it is new to me. Suppose I have registered for the test on 15th of November and the score will be received by the college I apply on 1st of December. But my application packages (CV, statement of purpose, diploma etc) will only arrive there one month...

Homework Statement Find the value of: lim x approaches 0 of : (3x - sin 3x) / (x2 sin x) Homework Equations Trigonometry identity Limit properties No L'hopital rule The Attempt at a Solution I tried changing sin 3x to -4sin3x + 3 sin x but then I stuck. Is changing sin 3x correct way to...

Hi, I know that when you take this limit it is equal to e^-wo, but I was just wondering how you got there when taking the limit? lim w-->wo ((exp(w)-exp(wo))/(w-wo))^-1 = 1/e^wo w and wo are both two points within the same plane.

I recently came across a Wikipedia article about somebody's (?) law regarding limits on a moon's orbital radius because the sun's gravitational influence is greater than the planet's at some distance from the planet. As I recall, the law had two different names associated with it. In addition to...

I was thinking recently about the Casimir Effect and at what distance does the effect become negligible? Is there any relevance on the surface area (or difference in areas) between the plates (I personally held the opposite, that any point can be considered to have equivalent pressure as any...

Homework Statement How can you determine the following limit using rationalization? where x = h Homework EquationsThe Attempt at a Solution I attempted to multiply by the conjugate and cannot get the problem to work out. When doing this I noticed no terms canceled out and now I am stuck. We...

Homework Statement Solve: ##\displaystyle \lim_{x\rightarrow 2} \frac{\left | x^2 + 3x + 2 \right |}{x^2 - 4}##. Homework EquationsThe Attempt at a Solution I am trying to solve the following limit: ##\displaystyle \lim_{x\rightarrow 2} \frac{\left | x^2 + 3x + 2 \right |}{x^2 - 4} =...