Limit Definition and 999 Threads

Homework Statement Does the following limit exist: ##\displaystyle \lim_{(x,y) \rightarrow (0,0)} = \frac{\sqrt{x^2+y^2+xy^2}}{\sqrt{x^2+y^2}}##? Homework EquationsThe Attempt at a Solution So I am trying to evaluate the limit along several curves, such as y=x, y=0, y=x^2, and I keep getting...

This is a simple exercise from Spivak and I would like to make sure that my proof is sufficient as the proof given by Spivak is much longer and more elaborate. Homework Statement Prove that \lim_{x\to a} f(x) = \lim_{h\to 0} f(a + h) Homework EquationsThe Attempt at a Solution By the...

Compute the following limit: $$\lim_{n\to\infty}\left(\sqrt[n]{\int_0^1 \left(1+x^n\right)^n\,dx}\right)$$

I'm trying to verify that: \lim_{(x,y)\rightarrow (0,0)}\frac{\sin (x^{3}+y^{3})}{x+y^{2}}=0. 0<\sqrt{x^2+y^2}<\delta\rightarrow |\frac{\sin (x^{3}+y^{3})}{x+y^{2}}|<\epsilon 0\leq |\sin (x^{3}+y^{3})|\leq |(x^{3}+y^{3})|\leq |x|x^2+|y|y^2 |\frac{\sin (x^{3}+y^{3})}{x+y^{2}}|\leq \frac{...

Homework Statement f [/B] f(x)= x^2 +4 find the limit as x approaches 1, there is something wrong with the latex code but I don't know what. Limit $$\lim_{x\to 1} \frac{{f(x)}^4-{f(1)}^4}{x-1}$$ Homework Equations -methods for finding limits -factorising polynomials -possibly polynomial long...

Hello, I have tried the integral below with Mathematica and it gives me the following solution: ##\frac{d}{dc}\int_{z^{-1}(c)}^{1} z(x)dx = -\frac{c}{z'(z^{-1}(c))}## I am not quite sure where it gets it from...I think it can be separated and with differentiation the first part will be zero...

Homework Statement Prove that the limit is 1 as x approaches 0 for the function f(x) / x. Homework EquationsThe Attempt at a Solution I put the f out in front so I was left with f•Lim as (x → 0) (x/x) so I was left with f•lim(x→0) 1 so I used the limit property and was left with f•1. That was...

Homework Statement Find ##\lim_{x\to\infty} x(e^{1/x}-1)## Homework Equations ##\lim_{x\to\infty} \frac{f(x)}{g(x)} = \lim_{x\to\infty} \frac{f'(x)}{g'(x)}## The Attempt at a Solution I attempted to rewrite the function in terms of a ratio and then use L'Hopital's rule: ##\lim_{x\to\infty}...

In general I'm wondering if \lim_{x\to0} \left[\frac{d}{dy} \frac{d}{dx} f(x,y)\right] = \frac{d}{dy} \left[\lim_{x\to0} \frac{d}{dx} f(x,y)\right] holds true for all f(x,y). Thanks.

Homework Statement lim as z--> i , \frac{z^2-1}{z^2+1} The Attempt at a Solution [/B]When we plug in i we get -2/0, so we get division by 0, Does this mean the limit is infinity, I also tried approaching from z=x+i where x went to 0, you get the same answer, I also approached from...

Homework Statement limit (x -> 0 y -> 0) of xy/sin(x+y) Homework Equations None that come to mind but maybe Lopital's Rule The Attempt at a Solution I know that the limit does not exist but I am having trouble figuring out how to show that it does not using the line x=y gives x^2/sin(2x)...

In your opinion, how old is too old to consider a four-year math degree? Can a person who is middle age return to college to major in math even if the degree itself will not lead to a rewarding career?

I can't prove it and I've got it by some intuition because not many properties of superlogarithms are known. I don't think anyone can prove it but is there some way to at least check if it is correct. The limit is: $$\lim_{h\rightarrow0}slog_{[log_xx+h]}[log_{f(x)}f(x+h)]$$ where ##slog## is the...

Homework Statement ##\frac{e^x-1}{x}## Evaluate the limit of the expression as x approaches 0. Homework Equations 3. The Attempt at a Solution [/B] The question i have is more theoretical. I was able to solve this problem by expanding the expression into the talyor polynomial at ##x=0##. I...

Hello all, I have a complicated function: \[f(x)=\left ( e^{x}+x \right )^{^{\frac{1}{x}}}\] I need to find it's derivative and it's limit when x goes to infinity. As for the derivative, I thought maybe to use LN, so that I can get rid of the exponent, am I correct? How should I approach...

Homework Statement Hi I am stuck on a small algebra set in the weak limit theorem to recover Newtonian equations The text I am looking at: ##\frac{d^2x^i}{ds^2}+\Gamma^i_{tt}\frac{dt}{ds}\frac{dt}{ds}=0## (1) ##\Gamma^{i}_{tt}=-1/2 \eta^{ij}\partial_{j}h_{tt} ## (to first oder in the...

Homework Statement Suppose a_{n}=\frac{n^2-2n+1}{2n^2+4n-1} For each positive number \epsilon , find a number N such that: \mid a_{n} - L\mid < \epsilon whenever n > N. Homework EquationsThe Attempt at a Solution \mid \frac{n^2 -2n + 1} {2n^2+4n-1} - \frac{1} {2} \mid < \epsilon...

So, a black hole has infinite gravity that even light can't escape from it, my question is, the gravitational field of a black hole can even pull light into it, then it means it is even faster than light, if not, light can escape from it. Does this argument make any sense, please tell me! thanks

I am given an implicit expression for an algebraic function, ##w(z)## as:##f(z,w)=(1-3 z+3 z^2-z^3)+(-4+8 z-4 z^2)w+(6-6 z)w^2+(-4)w^3+(1)w^4=0## with ##\displaystyle \frac{dw}{dz}=-\frac{\frac{df}{dz}}{\frac{df}{dw}}=\frac{6 w^2+8 w z-8 w+3 z^2-6 z+3}{4 w^3-12 w^2-12 w z+12 w-4 z^2+8 z-4}##...

It is said that the physical vacuum, is by definition a state with no "physical particles" -- more precisely, it is the ground state (state of lowest energy) of the field. Is there any beyond standard model where the vacuum is a lower energy limit of another theory or stuff... like the...

Homework Statement For which value of d does the following limit exist? lim x->d ln [ (x2-13x+30) / (x-d) ] Homework Equations None The Attempt at a Solution I understand how to find limits when the limit goes to a real number, and has a variable in the function to solve for, but not when...

in the limit definition of a sequence, why does epsilon has to be greater than 0 and not greater or equal to 0? thanks in advance.

Hello I am trying to solve this limit here: \[\lim_{x\rightarrow -\infty }\sqrt{x^{2}+3}+x\] I understand that it should be 0 since the power and square root cancel each other, while the power turned the minus into plus, and then when I add infinity I get 0. This is logic, I wish to know how...

In this solution, in the last 3rd line, I get the first part (-e^-1 - e^-1), however, after the '-' symbol, the person writes (1/b * e^1/b - e^1/b) and takes the limit as b->0. However, shouldn't this give him (inf. * e^inf - e^inf)? Thanks (His answer is correct, by the way)

We kept getting some voltage spikes back into our amplifier. One of the things we put in place is a 10kOhm resistor (discharge load) connected to the output of the amplifier. However, the resistor keeps burning and setting on fire. The amp company offered a way on checking this... I'm really...

Homework Statement I would like to understand how the limit was changed in the ratio test from step 1 to step 2 in the image that I've posted. I thought that the denominator would look like (2/n+2)(2/n+1) in step 2 since it looks like we are just turning the n's into reciprocals. Any help here...

Homework Statement Using the taylor series at point ##(x=0)## also known as the meclaurin series find the limit of the expression: $$L=\lim_{x \rightarrow 0} \frac{1}{x}\left(\frac{1}{x}-\frac{cosx}{sinx}\right)$$ Homework Equations 3. The Attempt at a Solution [/B] ##L=\lim_{x \rightarrow 0}...

Homework Statement the limit x->a of [(x+2)^5/3 - (a+2)^5/3] / (x-a) Homework Equations limit laws The Attempt at a Solution [/B]

Homework Statement $$\lim_{x\to\infty} \left(\frac{n^2+2n+1}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}$$ Homework Equations 3. The Attempt at a Solution [/B] I tried ##\lim_{x\to\infty} \left(\frac{n^2+2n+3-2}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}=## ##\lim_{x\to\infty}...

Evaluate $\displaystyle \lim_{n\rightarrow \infty}\prod^{n}_{k=1}\left(1+\frac{1}{4k^2-1}\right)$

Homework Statement Prove that $$ \lim_{x\to 0} \sqrt{x^3+x^2}\; \sin\left(\frac{\pi}{x}\right) = 0 $$ using Sandwich theorem Homework Equations Sandwich Theorem The Attempt at a Solution Now we know that sine function takes values between -1 and 1. ## -1 \leqslant...

Some claim not possible naturally, ie beat 2 hours. While not possible under race conditions Nike is trying to see if it is humanly possible; http://running.competitor.com/2016/12/news/nike-launches-program-break-2-hour-marathon-barrier-2017_160032

Basically i need to evaluate the limit of this expression ##\lim \sqrt[3]{n^6-n^4+5}-n^2=?## I want to know if this is correct and why: ##\lim \sqrt[3]{n^6-n^4+5}-n^2=\lim...

is there any limit to the expansion of the universe? (Cosmological Expansion) Can anyone refer to some research papers on this topic?

What does it mean when a limit is finite

Hello all. I am working on proving some theorems about Monte Carlo simulation and have proven a theorem that, in a certain formula, it is valid to replace a random variable in the denominator of a fraction by its expected value. I have been wondering whether this result can be generalised to...

Hey! :o I want to calculate the limit $$\lim_{x\rightarrow \infty}x^{100}\left [\frac{1}{x}\right ]$$ When $x\rightarrow +\infty$ it holds that $0<\frac{1}{x}<1$, or not? (Wondering) If yes, it holds that $\left [\frac{1}{x}\right ]=0$ or not? Then $x^{100}\left [\frac{1}{x}\right ]=0$, and...

Homework Statement I was sick and missed the lecture so having hard time with this problem @_@. http://i.imgur.com/ByK7iVk.png Homework Equations I don't know how to solve it all the textbook don't have particular problem so having a hard time figuring it out. The Attempt at a Solution My...

I have $\lim_{{x}\to{0}} lnx \cdot x$ $x$ approaches 0, and $lnx$ approaches $\infty$. How can I reason about this. I suppose $x$ approaches 0 more quickly than $lnx$ approaches $\infty$ , therefore it is zero. Is this accurate? How can I prove this.

Here's a graph and its triple integral. How are the limits of integration for the outer integral [-2,2]? I have no idea how this was found. Any help would be appreciated!

Mods, I wasn't sure whether to put this in quantum physics or relativity, but since the speed of light is the limiting factor I chose here. Move wherever you think is best. Okay so the speed of light is the asymptote for the speed that objects can accelerate to, and the Planck time is the...

In Schutz says When we have weak gravitaional fields then the line element *ds* is $$ ds^{2}=-(1+2\phi)dt^{2}+(1-2\phi)(dx^{2}+dy^{2}+dz^{2}) $$ so the metric is $$ {g_{\alpha\beta}} =\eta_{\alpha\beta}+h_{\alpha\beta}= \left( \begin{array}{cccc} -(1+2\phi) & 0 & 0 & 0\\ 0 & (1-2\phi) & 0 &...

Homework Statement http://prntscr.com/dcfe0u Homework EquationsThe Attempt at a Solution So I'm not really strong in proofs but I think you may be able to do something like this:$$lnL = \frac{ln(1+1/x)}{x}$$ $$lnL = \frac{1/x^2}{1+1/x}$$ and then more simplifying I get something like: $$lnL =...

One way that people introduce the Hawking temperature of an event horizon, is by taking the near-horizon limit of the BH metric and then do a Wick rotation of the time coordinate. Then, the regularity of the metric requires that the Euclidean time to be periodic. But how can this give us the...

Hey! :o I want to find the $\lim\sup$, $\lim\inf$ and the limit points of the following sequences: 1. $a_n=(-1)^n\frac{3n+4}{n+1}$ 2. $a_n=\sqrt[n]{n+(-1)^nn}$ 3. $a_n=\left ( \frac{n+(-1)^n}{n}\right )^n$ 4. $a_n=(-1)^{\frac{n(n+1)}{2}}\sqrt[n]{1+\frac{1}{n}}$ I have done the following...

Hey! :o Let $(a_n)_{n=1}^{\infty}$ be a real sequence and let $(b_n)_{n=1}^{\infty}$ a sequence in the set of limit points of $(a_n)_{n=1}^{\infty}$, $L(a_n)$. There is also a $b_0\in \mathbb{R}$ with $b_n\rightarrow b_0$ for $n\rightarrow \infty$. I want to show that then $b_0\in L(a_n)$...

How could I calculate: $\displaystyle \lim_{n \rightarrow + \infty}{(\sqrt(n^2+n) - \sqrt[3](n^3+n^2))}$ Everything that I tried I always got infinite forms or 0 in denominator. I don't have any idea what else should I try here.

Then is potential energy infinite (potential energy=energy that can be created)

I've got another limit question here, but I don't suppose you use the same method as last time for sure (rational). I essentially got (infinity)^0 and just assumed that to be equal to 1 which made sense to me. However, in my textbook and notes, (infinity) to the power of 0 is one type of an...

Hi, I am having trouble with these kind of questions where we have to use L'Hospital's Rule. I took the ln of the function to get the x out of the exponent, and then followed the Rule by taking the derivative of the top and bottom (using a shortcut we learned: lim x --> infty f(x)g(x) = lim x...