Limits Definition and 1000 Threads

Homework Statement \lim\limits_{x \to 0} \left(\ln(1+x)\right)^x Homework Equations Maclaurin series: \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} + ... + (-1)^{r+1} \frac{x^r}{r} + ... The Attempt at a Solution We're considering vanishingly small x, so just taking the first term in the...

Homework Statement Find the following limit.Homework Equations The Attempt at a Solution I cannot apply L' Hopital rule because it does not apply to this question. Hence I have no idea how to approach to this question. Please give me some guidelines.

Hi Community, I have the following question: I have done basic solving of limits and also of Riemann sums but never had to do them in the same question. Would I be correct in saying that I need to solve for the Riemann sum first then take the limit of the integral? Cheers Nemo

Homework Statement What will be lim[2sin(x-1)/(x-1)], where x tends to 1? [ ] denotes greatest integer function. Homework Equations Can I directly solve it using the formula sinx/x =1 when x tends to 0 The Attempt at a Solution Okay so the quantity inside [ ] can be written as ——>>2...

So, I'm still struggling with limits a bit.. Today, I've tried solving two different exercises which look pretty much the same. I could solve the first one relatively easily: $$\lim_{{x}\to{+\infty}}\frac{\sqrt{4x^{2}-1}-x}{x-3}$$ I applied the usual steps and arrived to the expression...

Homework Statement I need to mathematically prove that the center angle(s) (labeled as "A" in the photo below) approach what I believe to be 60 degrees (but never reach 60 degrees). We are given the values of all longer legs of each right triangle. Furthermore, the value of the length of each...

Hello all, I am trying to calculate the following limits, without cheating and using a calculator (by setting a very close value of the required value of x). And no l'hopital's rule either if possible :-) The limits are: \[\lim_{x\rightarrow 0} \frac{ln(x^{2}+e^{x})}{ln(x^{4}+e^{2x})}\]...

Hi everybody! I'm currently preparing a math exam, and I'd like to clear up a few points I find obscure about limits of sequences, my goal being to more or less determine a method to solve them quickly during the exam. Hopefully someone can help me here :) I'll number the questions so that it's...

Homework Statement Determine which of the sequences converge or diverge. Find the limit of the convergent sequences. 1) {asubn}= [((n^2) + (-1)^n)] / [(4n^2)] Homework Equations [/B] a1=first term, a2=second term...an= nth term The Attempt at a Solution a) So I found the first couple of...

Homework Statement Problem is in image uploaded Homework Equations n/a The Attempt at a Solution x = u, y = v and z = 1 - u - v ∂r/∂u × ∂r/∂v = i + j + k F dot N = u^2 + 3v^2 ∫∫(u^2 + 3v^2 )dudv My problem is I'm not sure what I should take as the limits? Should I flip around the order of...

Hi I was wondering how you get this when taking the limit of T going to 0 From this expression of S: Please help I don't see how ln infinity goes to uB/KbT (used u to represent the greek letter. And how does the other expression of sinh and cosh approach 1?

Homework Statement I am supposed to determine whether the summation attached is convergent or divergent Homework Equations Alternating Series Test Test for Divergence The Attempt at a Solution The attempted solution is attached. Using the two different tests I am getting two different answers.

Homework Statement find the limit of arccoshx - ln x as x -> infinity Homework Equations ##arccosh x = \ln (x +\sqrt[]{x^2-1} )## The Attempt at a Solution ## \lim_{x \to \infty }(\ln (x + \sqrt{x^2-1} ) - \ln (x)) = \lim_{x \to \infty} \ln (\frac{x+\sqrt{x^2-1}}{x}) \ln (1 + \lim_{x \to...

Homework Statement I have to determine whether or not the following sequence is convergent, and if it is convergent, I have to find the limit. an = (-2)n / (n!) In solving this problem, I am not allowed to use any form or variation of the Ratio Test. 2. The attempt at a solution I was...

Homework Statement When we are taking a limit of a multivariable function, why do we use the points (x,0) and (0,y) to find out if the limit doesn't exist? They are two different points, no? If they are two different points then wouldn't they go to different points on the z-axis? If we have a...

Homework Statement Evaluate or show that the limit does not exist: \lim_{(x,y) \to (0,0)}\frac{ 2x^{4} + 5y^{3} }{8x^{2}-9y^{3}} \lim_{(x,y) \to (0,-2)}\frac{ xy+2x }{3x^{2}+(y+2)^{2}} Homework EquationsThe Attempt at a Solution So the first one is indeterminate and cannot be factored...

Hello! (Wave) I want to find the following limit, if it exists. $\lim_{(x,y) \to (0,0)} \frac{\cos x-1-\frac{x^2}{2}}{x^4+y^4}$ If we say : let $(x,y) \to (0,0)$ along the line $y=0$ , what exactly does it mean geometrically? Also, if we want to check whether the limit $\lim_{(x,y) \to...

I'm trying to calculate this limit to answer a question in Quantum Mechanics: \mathop {\lim }\limits_{{t_1} \to 0} \,\,{\left( {\frac{m}{{2\pi \hbar i{t_1}}}} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}{e^{im{{(x' - x)}^2}/2\hbar...

Homework Statement I need to see if the function defined as##f(x,y) = \left\{ \begin{array}{lr} \frac{xy^2}{x^2 + y^2} & (x,y)\neq{}(0,0)\\ 0 & (x,y)=(0,0) \end{array} \right.## is differentiable at (0,0) Homework Equations [/B] A function is differentiable at a...

Hi, I'm having some trouble with finding the limit for this question: I can use the l'hopital's rule which I tried.. I tried pi, 2pi, 0, inf, none seem to work so if I could have some help that would be appreciated! limx→0 $$\frac{cos5x-cos6x}{x^2}$$ Would the degree rule apply here? It...

I was looking over my lectures notes and became stumped when the lecturer began manipulating an integral using a dummy variable. I've attached the integral below as a JPEG. I can follow the substitution up until the very end but am having trouble understanding why he replaced u with a positive s...

I am a noobie and I am not really sure this is the right forum to post to, so please excuse me if I am posting in the wrong place. I had considered General Physics and Classical Physics, but I hope I have made my best choice and I don't think it is acceptable to post in more than one forum...

Homework Statement Find the following limits, if they exist. Homework EquationsThe Attempt at a Solution I have just started calculus and am having trouble with 3 a). I get 0/0 after substitution so I factored but still get 0 in the denominator. Does this indicate that the limit does not...

I've been looking at detector coincidences and tried to find what general limits apply to coincidences. I was surprised how simply the calculation works out. My question is whether it is correct and where can I find similar stuff ? Consider two binary sequences produced by random processes...

I would like to post a comment for offtopic conversation in another thread. This is the point of no-communication theorem that measuring one particle does not change anything measurable about the other particle. But conclusions of no-communication theorem are limited by it's assumptions (as for...

hi I don't understand how to do one type of homework problem, here's an example of the type: If limit of f(X)/X = 1 as X ->0 evaluate the limit f(X) as X->0

Homework Statement Use algebra to show that U(x) = −√x − 1 and L(x) = −√x satisfy the ’funnel condition’ U(x) − L(x) → 0 as x → ∞ Homework Equations Funnel condition: The two fences come together asymptotically, i.e. U(x) − L(x) is small for large x. The Attempt at a Solution I think that...

Is it true that every limit that takes on an indeterminate form can be evaluated? Is it proper to say that a limit problem has a solution if the limit does not exist?

Homework Statement f(x)=-2 when x<1 =3 when x=1 =x-3 when x>1 find the limit at 1 from the left and right sides and at 1. Homework EquationsThe Attempt at a Solution limit for x when approaching 1 from the left is -2 limit for x when approaching 1 from the right is -2 -I'm not sure...

Homework Statement If k is a positive integer, then show that ##\lim_{x\to\infty} (1+\frac{k}{x})^x = \lim_{x\to 0} (1+kx)^\frac{1}{x}## Homework Equations L'Hopitals rule, Taylor's expansion The Attempt at a Solution How should I begin? Should I prove that both has the same limit, or is...

Homework Statement The picture attached appeared in my powerpoint for my class. It's been a long time since I took calculus 1, but if I remember correctly this formula is wrong correct? I mean thinking about it limit k-> inf ( cos(theta)^k ) = 0 if theta is not a multiple of pi OR +/- 1 if...

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...

I have read Spivak's Calculus up to chapter 5, which is on Limits. Up until this point, the majority has been very straightforward and easy to understand. However, I am having trouble grasping the concept of limits in the style/method that Spivak describes them. Can anyone elaborate in a more...

Homework Statement Given that ##T = \frac{4E(V-E)}{4E(V-E)+V^2\sinh^2 (ka)}## Simplify the expression for T when a is large but not infinite, and again for the case when a tends to zero and the potential tends to infinity, such that ##d = Va## is a real constant. k is a constant, as is E...

I have been playing around with calculus for a while and I wondered what would it be like to make some changes to the definition of derivatives. I'd like to look at the original definition of derivatives in this way (everything is in lim Δx→0): F(x+Δx) - F(x) = F'(x) * Δx The Δx factor...

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...

I am using Spivak calculus. The reason why epsilon-delta definition works is for every ε>0, we can find some δ>0 for which definition of limit holds. Spivak asserts yhat if we can find a δ>0 for every ε>0, then we can find some δ1 if ε equals ε/2. How is this statement possible? Since ε>0, then...

Consider the limit lim f(x)g(x) x→a Spivak has proved that this is equal to lim f(x) multlied by x→a lim g(x) x→a And also if lim g(x) = k and k≠0, x→a Then. lim 1/g(x) = 1/k x→a Now the...

I've seen in some lecture notes that the proper distance dp(t) can be written as ##\int_{t_e}^{t_0} c dt/a = \int_0^z c dz /H(z)## I can perform this integral ok using ##H =\dot a/a## and the fact that ##1 + z = 1/a(t_e)## but it requires associating the limits of the integration as te...

Find the Fourier Transform of $ e^{-a|t|}Cosbt $ I'd like to simplify this using $Cosbt = Re\left\{e^{ibt}\right\}$ $\therefore \hat{f}(\omega) = Re\left\{ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{\left(-a+ib+iw\right)|t|} \,dt \right\} = Re\left\{ \frac{1}{\sqrt{2\pi}}...

jfizzix submitted a new PF Insights post Understanding Buoyancy: The Limits of Archimedes Continue reading the Original PF Insights Post.

I have a hammock that is rated to 400 lbs. I want to use two neodymium magnets to suspend the hammock from two steel posts. Do I need two magnets rated 400 lbs each?

hello, sorry for bad English, i have a question. if we consider the following equations and we take natural values note that tend 2 x-1=0 -----------------> x = 1 x^2-x-1=0 ----------------->...

İn a finite set, can we take limit to $\infty$ ? Also, can you give an example related to infinite subset of $\Bbb{N}$ ?

Noting that ice expands by about 9%, why isn't it possible ot build a heat engine from this natural process?

Homework Statement From Spivak: Suppose that ##A_{n}## is, for each natural number ##n##, some finite set of numbers in ##[0,1]##, and that ##A_n## and ##A_m## have no members in common if ##m\neq n##. Define f as follows: ##f(x) = \frac{1}{n}##, if ##x \in A_n## ##f(x) = 0##, if ##x \notin...

Homework Statement If ## lim_{x\to 0^+} f(x) = A ## and ## lim_{x\to 0^-} f(x) = B ##, find: (a) ## lim_{x\to 0^+} f(x^3-x) ## (b) ## lim_{x\to 0^-} f(x^3-x) ## Homework Equations None. Only the problem statement. The Attempt at a Solution (a) ## lim_{x\to 0^+} f(x^3-x) = lim_{x\to 0^+}...

Simple question, simple example. (x-1) \frac{(x-2)(x-1)}{x-2} So, it's quite clear that the contents of the two lines are not equal to one another. Since \frac{1}{1}\neq\frac{x-2}{x-2}, how is it the case that we can treat it as \frac{1}{1}=\frac{x-2}{x-2} when we're doing limits? Edit: Oh...

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...