Limit Definition and 999 Threads

Homework Statement How $$ \lim_{x\to\infty} \frac{nlogx}{[x]} = 0 $$ ? Here n∈ ℕ Here [x] is greatest integer or floor function of x. Homework Equations [x] = x - {x} where {x} is fractional part of x. The Attempt at a Solution I know floor function is not differentiable. We are getting here...

Homework Statement Calculate the following limit if it exists ## \lim_{z\to -1}\frac{\sqrt{z}-i+\sqrt{z+1}}{\sqrt{z^2-1}} ## the branch of root is chosen so that ## \sqrt{-1}=i## Homework EquationsThe Attempt at a Solution I tried most of the same things that I tried earlier today (...

Homework Statement Calculate the following limit if it exists: ##\lim_{z\to i} = \frac{z^3+i}{z-i}## Homework Equations Possibly relevant: ## \lim_{z\to\infty} f(z) = \omega_0 \hspace{5mm} \text{if} \hspace{5mm} \lim_{z\to 0} f\left(\frac{1}{z}\right) = \omega_0## The Attempt at a Solution...

Hi, I need help with the following limit, the solution is apparently -3/2 but I don't get it. Question: limit as n approaches infinity of [ sqrt(n^2 +3n - 4) - sqrt(n^2 + 6n +5) ] Attempt: So I was just thinking to factor out n like: (n^2)^(1/2) (1 + 3/n - 4/n^2)^1/2 - n (n^2)^(1/2)...

Homework Statement $$ \lim_{x\to\infty} (\sqrt{a^2x^2 +bx + x} -ax) = $$ b/2a b/a 0 2b/a Homework Equations Lim x tends to infinity 1/x = 0 The Attempt at a Solution Taking a2x2 common from square root, we get $$ \lim_{x\to\infty} (ax \sqrt{1+\frac{b}{a^2x}+ \frac{1}{a^2x}} -ax) = $$...

Hello, I was just wondering how to solve this limit: Limit of (3/4)^(n+1) as n approaches infinity My attempt: (3/4)^(n+1) = (3^ (n+1) ) / (4 ^ (n+1)) Top goes to infinity and bottom goes to infinity to use l'hopital rule. lim = ( ln(3) * 1 * 3^(n+1) ) / ( ln(4) * 1 * 4^(n+1) ) But the...

Hello, I was just wondering how to solve this limit: Limit of (3/4)^(n+1) as n approaches infinity My attempt: (3/4)^(n+1) = (3^ (n+1) ) / (4 ^ (n+1)) Top goes to infinity and bottom goes to infinity to use l'hopital rule. lim = ( ln(3) * 1 * 3^(n+1) ) / ( ln(4) * 1 * 4^(n+1) ) But the...

Having a question regarding Faraday's cage,Applying a magnetic field over the cage, will cause the charges in the conducting cage to rearrange, thus causing another field which opposes the first field. This will give a net field inside the cage of zero (this is why we are safe in a car during a...

What is the proof for this $$ \int_a^b f(x) dx = 1/n\lim_{n\to\infty} (f(a) + f(a+h) + f(a+2h) +...+ f( a+ (n-1)h)) $$ h = (b-a)/n Also I think there is some summation form which can be converted to integral form how?

Homework Statement The function f is defined f(x)=floor(x^2)/x^2 I need to find the limit of the function at an arbitrary point. For the continuous parts it was fine, and also for right sided limit at positive points of discontinuity (and left sided for negatives, for all of which the lim is...

Homework Statement [/B] ##\lim x\rightarrow \frac{\pi }{4} (\frac{1-\tan x}{\sin x - \cos x})## The Attempt at a Solution [/B] By assuming y = x-π/4 , the limit become : ## \lim y\rightarrow 0 (\frac{1- \tan (y+\frac{\pi}{4})}{\sin (y+\frac{\pi}{4}) - \cos (y+\frac{\pi}{4})}) = \lim...

Homework Statement lim x->∞ (2^x-5^x) / (3^x+5^x) Choices : a. -1 b. -2/3 c. 1 d. 6 e. 25 2. The attempt at a solution Hmmm.. I really have no idea about this.. This is an unusual problem.. Please tell me...

Determine $$\lim_{{n}\to{\infty}}\frac{(n!)^{1/n}}{n}$$

Using epsilon delta, prove $$\lim_{{n}\to{\infty}}\frac{2^n}{n!}=0$$ Doesn't seem too difficult, but I have forgotten how to do it. Obvious starting point is $\forall \epsilon >0$, $\exists N$ such that whenever $n>N,\left|\frac{2^n}{n!} \right|<\epsilon$.

Homework Statement I'm reading a derivation and there is a step where the writer goes from: ## \sum_{n=0}^\infty e^{-n\beta E_0}## to: ## \frac {1} {(1-e^{-\beta E_0})}.## I can't see how they did this.Homework Equations [/B] I think it just involves equation manipulation. The Attempt at...

Homework Statement Find the limit of : lim x-> (π/2) (2-2sin x)/(6x-3π) 2. The attempt at a solution lim x-> (π/2) (2-2sin x)/(6x-3π) =lim x-> (π/2) 2-2 sin x / 6 (x- (1/2)pi) Assuming that y = x - (π/2) So, lim y->0 (2-2sin(y+pi/2))/6y lim y->0 (2-2 (sin y cos pi/2 + cos y sin pi/2)/6y...

Homework Statement It is not exactly a homework question, but why does the definition of a limit use strict inequalities as follows: if 0 < |x - a| < δ, then |f(x) - l| < ε rather than weak inequalities, for example if 0 < |x - a| < δ, then |f(x) - l| ≤ ε Could the addition of the equality...

How can it be proved that as lim n tends to infinity, (n2-1)/(n2 + n + 1) tends to 1 ?

Homework Statement Determine whether the sequence converges or diverges. If it converges, find the limit. Here's the sequence: http://www4a.wolframalpha.com/Calculate/MSP/MSP89541ea2ag9dg617bcd6000050d52e94i67ei593?MSPStoreType=image/gif&s=39&w=66.&h=44. Homework Equations N/A The Attempt at...

Homework Statement Assume five hundred people are given one question to answer - the question can be answered with a yes or no. Let p =the fraction of the population that answers yes. Give an estimate for the probability that the percent of yes answers in the five hundred person sample is...

I need help with the following theorem: Let I, J ⊆ℝ be open intervals, let x∈I, let g: I\{x}→ℝ and f: J→ℝ be functions with g[I\{x}]⊆J and Limz→xg(x)=L∈J. Assume that limy→L f(y) exists and that g[I\{x}]⊆J\{g(x)},or, in case g(x)∈g[I\{x}] that limy→L f(y)=f(L). Then f(g(x)) converges at x, and...

The hint I found in http://math.stackexchange.com/questions/448207/how-to-prove-that-lim-limits-x-to0-frac-tan-xx-1#answer-448210 limx→0(tan(x) / x)= limx→0( (tan(x)−0) / (x−0)) = limx→0 ( (tan(x)−tan(0) ) / (x−0) )=⋯ Then, I don't know how to continue it.. What identity is used ? I don't see...

Homework Statement Let f_1,f_2\colon\mathbb{R}^m\to\mathbb{R} and a cluster point P_0\in D\subset\mathbb{R}^m (domain) Prove that \lim_{P\to P_0} f_1(P)\cdot f_2(P) = \lim_{P\to P_0} f_1(P)\cdot\lim_{P\to P_0} f_2(P) Homework EquationsThe Attempt at a Solution Let \begin{cases} \lim_{P\to...

Homework Statement I'm struggling with the proof that the limit of a complex function is unique. I'm struggling to see how |L-f(z*)| + |f(z*) - l'| < ε + ε is obtained. Homework Equations 0 < |z-z0| < δ implies |f(z) - L| < ε, where L is the limit of f(z) as z→z0 .The Attempt at a Solution...

Assume we have a free-falling particle in gravity in a static metric. Its worldline is described by: g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} where ##|h_{\mu \nu} << 1|##. Taken from Hobson's book: Why did they let ##g^{k\mu} = \eta^{k\mu}##?

Prove that the limit as n approaches infinity of ((2^n * n!)/n^n) equals to zero. The hint is to use Stirling's approximation. What is this?

In his 2001 Three Roads to Quantum Gravity, on its p.l66, Smolin says, "M theory, if it exists, cannot describe a world in which space is continuous and one can pack an infinite amount of information into any volume, no matter how small." As a lay person, I'm hoping to get an informed opinion...

Hey! :o I have found applying De L'Hoptal's Rule that $$\lim_{x \rightarrow 0} \frac{\sin 2x-2x}{x^3}=-\frac{4}{3}$$ Now I am asked whether the limit $$\lim_{(x, y) \rightarrow (0, 0)} \frac{\sin 2x-2x+y}{x^3+y}$$ or not. How could we check that ?? (Wondering)

Homework Statement (hebrew) : f(x) a continuous function. proof the following Homework Equations I guess rules of limits and integrals The Attempt at a Solution I've tried several approaches: taking ln() of both sides and using L'Hospitale Rule. Thought about using integral reduction...

I seem to have two approaches that I've seen and understand, but I can't quite see how they relate. 1. Write a general time evolving state as a superposition of stationary states multiplied by their exp(-iEt/h) factors, and calculate <x>. We find that <x>=Acos(wt+b) as in classical physics (in...

I have seen light pipes which "conduct" sunlight from an outdoor receptor to the dark recesses of buildings ... but they all seem to be awkward bulky metallised tubes. Is it not possible to achieve the same effect with fibre optics? Could one not focus the light at the receptor into a...

I try to figure it out but I can't get the answer that I need and when I look upon the solution from the book I don't understand it at all. The answer is " no limit" and there is no explanation why. The question is Determine the limit of lim (x2+y2)- -> infinity (xye-(x+y)2 in this case I use...

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...

We see always in source meter machines a LED which indicates the limit current. I want to know what is the limit current and what is the relationship between this later and the resistivity.

I would really appreciate if you could help me solving this limit problem! Determine the limit without using L'Hospital's rule! $$ \lim_{x\to -2} \sin(\frac{\pi x}{2})\frac{x^2+1}{x+2} = ?$$ Thank you in advance!

Let x(a) be the extinction probability of a branching process whose offspring is Poisson distributed with parameter a. I need to find the limit as a approaches infinity x(a)e^a. I tried computing x(a) directly using generating functions, and I found that it's the solution to e^(a(s-1))=s, but...

Homework Statement See attached image. Homework Equations Left- and right-handed limits. The Attempt at a Solution I know lt (t -> 12-) f = 150 mg and lt (t -> 12+) f = 300 mg, but I don't know how to explain these numbers. I'm assuming that measurements are taken and that this graph is...

Homework Statement Integral of ∫1/x^2 (or ∫x^-2) between 1 and 0.The Attempt at a Solution I can integrate it no problem to give me -1/x or x^-1, but when I put it between the limits of 1 and 0 I get ∞-1 which is just ∞. Is this right or do I need to use L'Hopital's rule. If so, how? I'm...

Why is the limit not just infinity? wouldn't it be (1-infinity)/(1+infinity)?

Hey everyone! So I have that the low temperature limit of a paramagnet is Ω=(Ne/Ndown)Ndown while the low temperature limit of an einstein solid is Ω=(Ne/q)q. How could I explain that these two equations are essentially the same considering their respective limits (Ndown<<N and q<<N) and that...

Is there a limit to the amount of energy which can be extracted from the wind? There are a huge number of windfarms springing up around the world.. all taking energy from the wind. The assumption seems to be that this is limitless and "free". Clearly this is not possible. The question is (I...

How do I justify that lim_{(x,y)\to (0,0)} \cos{\frac{x}{\sqrt{y}}} = 1? If I approach from the y axis, it would become lim_{y\to 0} \cos{\frac{0}{\sqrt{y}}} = 1 , but if I approach from the x axis, it would become lim_{x\to 0} \cos{\frac{x}{\sqrt{0}}} = D.N.E, no? (does not exist) Wolfram...

Hello guys , have a question and I can not find the answer. In what units is measured elastic limit ? Thanks A lot !

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...

Suppose $f$ is a continuous function on $(-\infty,\infty)$. Calculating the following in terms of $f$. $$\lim_{{x}\to{0}}f\left(\int_{0}^{\int_{0}^{x}f(y) \,dy} f(t)\,dt\right)$$

Homework Statement What is Lim (1+x2)/(4-x) as x approaches 4 from the left? Prove using the definition. Homework EquationsThe Attempt at a Solution Well x≠4. Function approaches positive infinity as x approaches 4 from the left side. Let m>0 and 0<x<4. Then (1+x2)/(4-x) > x2/(4-x) > x/(4-x) >...

Homework Statement lim x→4 √x-4 I need to do something so that it is not undefined or 0. Homework EquationsThe Attempt at a Solution I tried rationalizing, but that just gave me x-4/√x+4, which would still result in an undefined answer.

Hi. My nephew asked me a good question. I am trying to understand the Planck limit. It is said that we can not go lower than the Planck limit. But if we had a an imaginary powerful microscope to see at the plank level, and if we placed 2 Planck end to end with "half" a Planck sized length...

Hey MHB ! I've got a question that I am clueless how to proceed Prove that $$\Large \lim_{(x,y)\to (0,0)}(1+x^2y^2) ^{\frac{-1}{x^2+y^2}} = 1$$ Any hint would be appreciated.

Hello! (Wave) I am looking at the following exercise: Let the (linear) differential equation $y'+ay=b(x)$ where $a>0, b$ continuous on $[0,+\infty)$ and $\lim_{x \to +\infty} b(x)=l \in \mathbb{R}$. Show that each solution of the differential equation goes to $\frac{l}{a}$ while $x \to...