Limit Definition and 999 Threads

I need to find the limit of $$\left| \frac{(n + 1)n^3}{(n + 1)^{3}} \right|$$ as $n$ approaches infinity. I simplify this to: $$\left| \frac{n^3}{(n + 1)^{2}} \right|$$ But the solution simplifies it to: $$\left| \frac{n}{(1 + \frac{1}{n})^{2}} \right|$$ How do I get to this result?

Homework Statement I have the function: f(x,y)=x-y+2x^3/(x^2+y^2) when (x,y) is not equal to (0,0). Otherwise, f(x,y)=0. I need to find the partial derivatives at (0,0). With the use of the definition of the partial derivative as a limit, I get df/dx(0,0)=3 and df/dy(0,0)=-1. However, my...

I've pondered at this problem for a long time, and I don't know where I make a mistake, can anyone give me a hint? 1. Homework Statement find a ##\delta## such that ##|f(x)-L|<\epsilon## for all x satisfying ##0<|x-a|<\delta## Homework Equations [/B] ##f(x)=x^4##; ##l=a^4##The Attempt at a...

Homework Statement limit (5/(2+(9+x)^(0.5))^(cosecx) x-->0 attempt: tried applying lim (1+x)^(1/x) = e. x->0 couldn't get anywhere.

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{1}{\binom{n}{k}}$ is I have tried like this way:: Let $\displaystyle a_{n} = \sum^{n}_{k=0}\frac{1}{\binom{n}{k}} =...

I am trying to learn all the methods of finding the limit of a multivariable function. If I have $$\lim_{{(x, y)}\to{(0,0)}} \frac{x}{x^2 + y^2}$$ I can set $y = mx$ to see if the function solely depends on $m$, in which case the limit does not exist. So I would get $$\lim_{{(x...

Allo, When I was experimenting with graphing functions, I noticed the inverse tangent, or arctanget, curves away from y=2, or may be less. What is the y limit for the inverse tangent function? Does it for ever increase, or terminate at a co-ordinate?

I need to find $$\lim_{{(x, y)}\to{(0,0)}} \frac{x^2 - y^2}{\sqrt{x^2 + y^2}}$$ If I plug in zero, I get an indeterminate form. How do I resolve the indeterminate form?

I have $$\lim_{{(x, y)}\to{(0, 0)}} \frac{x^4 - y^4}{x^4 + x^2y^2 + y^4}$$ If I evaluate the limit along the x-axis, I get $$\lim_{{(x, y)}\to{(0, 0)}} \frac{x^4 - y^4}{x^4 + x^2y^2 + y^4}$$ which evaluates to $1$. If I evaluate the limit along the y-axis, I get $$\lim_{{y}\to{0}} \frac{...

I have $$\lim_{{(x, y)}\to{(0, 0)}} \frac{x}{x^2 + y^2}$$ We can approach the limit on the x-axis, so the values of $x$ will change and the values of $y$ will stay : $$\lim_{{x}\to{0}} \frac{x}{x^2}$$ I suppose I can take hospital's rule and get $$\lim_{{x}\to{0}} \frac{x}{x^2}$$...

I have $$\lim_{{n}\to{\infty}} \frac{|x|^2}{(2n + 3)(2n + 2)}$$ I can see that for smaller values of $x$ the limit is 0, but what if $x$ equals infinity, wouldn't that be an indeterminate form?

The definition of a limit of a sequence, if the limit is finite, is: lim n >infinity un (un is a sequence) = l <=> ∀ε> 0, ∃N: n > N => |un - l| < ε This just means that un for n > N has to be a number for which: l -ε < un < l + ε Now, I'm wondering, can't we just say: n > N => |un -l| <...

Hello! Could anybody help me? My wondering seems so trivial, but I can't skip it. They say that since u and d quarks are much lighter than QCD scale(~200MeV), in reality we can consider the QCD Lagrangian has an approximate global chiral symmetry with respect to these two flavors. At first, it...

The limit comparison test states that if $a_n$ and $b_n$ are both positive and $L = \lim_{{n}\to{\infty} } \frac{a_n}{b_n} > 0$ then $\sum_{}^{} a_n$ will converge if $\sum_{}^{} b_n$ and $\sum_{}^{} a_n$ will diverge if $\sum_{}^{} b_n$ diverges. Does this rule also apply if $L$ diverges to...

I'm trying to find the limit of this function: $$\lim_{{n}\to{\infty}} \frac{n^n}{({n + 1})^{n + 1}}$$ I can simplify it to this: $$\lim_{{n}\to{\infty}} \frac{n^n}{({n + 1})^{n}(n + 1)}$$ But I'm not sure of the best way to proceed.

The problem $$ \lim_{x \rightarrow \infty} \left( \sqrt{x+1} - \sqrt{x} \right) $$ The attempt ## \left( \sqrt{x+1} - \sqrt{x} \right) = \frac{\left( \sqrt{x+1} - \sqrt{x} \right)\left( \sqrt{x+1} + \sqrt{x} \right) }{\left( \sqrt{x+1} + \sqrt{x} \right) } = \frac{x+1 - x }{\left(...

Homework Statement Mod note: Edited the following to fix the LaTeX[/B] compute ##\lim_{n \rightarrow +0} \frac {8-9cos x+cos 3x} {sin^4(2x)}####\lim_{n \rightarrow +\infty} \frac {\sin(x)} x## ##\lim_{n \rightarrow +\infty} \frac {\sin(x)} x##ok find limit as x→0 for the function ##[ 8-9cos x...

The problem \lim_{x\rightarrow \infty} \frac{x^4 + x \ln x}{x + \left( \frac{2}{3} \right)^x} The attempt \lim_{x\rightarrow \infty} \frac{x^4 + x \ln x}{x + \left( \frac{2}{3} \right)^x} = \lim_{x\rightarrow \infty} \frac{x^4(1 + \frac{x \ln x}{x^4}) }{x + \left( \frac{2}{3} \right)^x}...

I have this limit: $$\sum_{k = 1}^{\infty} {(\frac{e }{3})}^{k}$$ Which method can I use to find if it converges or diverges?

We are starting sequences, and in one of the examples we have this limit: $$\lim_{{n}\to{\infty}} \frac{R^n}{n!}$$ We let $M$ equal a non-negative integer such that $ M \le R < M + 1$ I don't get the following step: For $n > M$, we write $Rn/n!$ as a product of n factors: $$\frac{R^n}{n!}...

I have this limit: $$\lim_{{n}\to{\infty}} {(\frac{2}{3})}^{n}$$ I know the answer is 0 but how can I evaluate this?

I have this limit: $$\lim_{{x}\to{\infty}} {(-1)}^{n}{n}^{3} + {2}^{-n}$$ and I'm unsure how to evaluate it or how to apply L'hopital's rule to this limit.

If I have this sequence $$a_n = \ln\left({\frac{n}{n^2 + 1}}\right)$$ I need to find: $$ \lim_{{n}\to{\infty}} \ln\left({\frac{n}{n^2 + 1}}\right)$$ Shouldn't I be able to find the limit of$$ \lim_{{n}\to{\infty}} \frac{n}{n^2 + 1}$$ (which is $0$) and then substitute the result of that...

I have this sequence: $${a}_{n} = \ln \left(\frac{12n + 2}{-9 + 4n}\right)$$ I need to find the limit of this sequence. How can I go about this? Do I need to apply L'Hopitals rule? I'm unsure how to simplify this expression. If I use the rule $\ln(\frac{a}{b}) = \ln a - \ln b$ I get $\infty -...

Hey! Can somebody take a look on these two limit problems? I don't agree with the answer to #15, which is supposed to be 0 while I get infinity. #16 seems to ask to find the value of the sum...I posted a pic of my attempts to solve the problems below. My attempts:

lim_(h->0^-) (e^(x+h)/((x+h)^2-1)-e^(x+h)/(x^2-1))/h = -(2 e^x x)/(x^2-1)^2 I know how to differentiate the expression using the quotient rule; however, I want to use the limit definition of a derivative to practice it more.This desire to practice led me into a trap! Now I just can't simplify...

How can you compute F(k) = k\int^{\infty}_{0}dy\int^{y}_{0}dx f(kx,y) in C. I know about Python's scipy.integrate.dblquad function but it's just too slow. I have written some Cython code with a 2D gaussian quadrature function in C but it only takes doubles as limits. I think C doesn't have...

Dr. Achim Rosch, a theoretical physicist at the University of Cologne in Germany, who proposed the technique used by Dr. Ulrich Schneider and his team to create in laboratory negative absolute temperature, have calculated that whereas clouds of atoms would normally be pulled downwards by...

If I have this limit: $$\lim_{{R}\to{1}} \frac{1}{R - 1}$$ I try to apply L'hopital's rule: The derivative of 1 is 0, and the derivative of $R - 1$ is 1. So I get $\frac{0}{1}$ which is 0. But apparently the answer is infinity. What am I doing wrong?

Homework Statement I am posting this for another student who I noticed did not have the proof in the problem. Here is what she said. Let's try and help her out. I have been working on the problem below and I am stuck. I am stuck primarily because of the part where is says x=0. If x-0, it...

Did not know how to word this properly. Looking for an equation to show the behaviour of how EM can be "lensed" as a function of wavelength/frequency. ie RF waves can not be lensed/focussed to a spot. what equation determines the minimum spot size a lens can focus EM radiation as a function...

Homework Statement Hey guys. I am having a little trouble answering this question. I am teaching myself calc 3 and am a little confused here (and thus can't ask a teacher). I need to find the limit as (x,y) approaches (0,1) of f(x,y) when f(x,y)=(xy-x)/(x^2+y^2-2y+1). Homework Equations...

Hello everyone. I need help trying to calculate/ trying to realize what the limit function of (sin nx)/(sin x) as n goes to infinity is. from another topic here on MBH ("Show δn = (sin nx) / (pi x) is a delta distribution") and after research with Wolfram Alpha I know that the limit function...

I have only encountered questions that f(x)-L that can be factorize to get a constant, and delta is epsilon divide that number, as a high school student. I have no idea how to choose a epsilon for this question. Thanks.

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...

this is a geometric proof from James Stewart's calculus textbook page 192. I'm confused in the sequence of inequalities as part of the proof... theta = arcAB < AB + EB ==> arcAB < AE + ED. How did EB turned into ED? please check the picture I've uploaded with this post

Homework Statement Section is on using power series to calculate functions, the problem is on proving the limit, solution is also attached but I do not see how the solution proves the limit. Homework Equations Convergent power series form The Attempt at a Solution I attempted to represent...

Hi All. I have a doubt concerning the limit: $$ \lim_{n \to \infty} \frac{\pi (n)}{Li(n)} = 1 $$. This mathematical statement does not imply that both functions converge to the same value. The main reason is that both tend to infinity as n tend to infinity. I would like to ask you if it is...

Homework Statement Hi. My professor asked me if I know to solve this limit and I tried doing it, however I didn't get the same answer as him. Question: What is the limit of (cos(x)*cos(2x)*cos(3x)*...*cos(nx)-1)/x^2 as x approches 0 Homework Equations / The Attempt at a Solution [/B] So to...

I know that Killing horizon is the hypersurface on which timelike Killing vector field becomes null. Beyond that surface Killing vector field becomes spacelike. But Stationary Limit Surface has also such a property. I wonder, if they are the same thing, if so, why is there different names for...

Homework Statement If a and b satisfy ##\lim_{x->0}\frac{\sqrt{ax+b}-5}{x} = \frac{1}{2}##, then a+b equals... A. -15 B. -5 C. 5 D. 15 E. 30 Homework Equations L'hospitalThe Attempt at a Solution By using L'hospital, I get b=a^2 Then, I got stuck.. Substituting b=a^2 into the limit...

Homework Statement Calculate ##\lim_{x\to 1} \frac{1}{x-1} \int_{1}^{f(x)} sin(\pi t^2) dt##. f is differentiable in the neighbourhood of point ##x=1## and ##f(1)=1##. Homework Equations If ##f## is continuous on a closed interval ##[a,b]##, then there exists ##ξ∈]a,b[## such that...

Homework Statement ##\lim_{a\rightarrow b} \frac{tan\ a - tan\ b}{1+(1-\frac{a}{b})\ tan\ a\ tan\ b - \frac{a}{b}}## = ...Homework Equations tan (a - b) = (tan a - tan b)/(1+tan a tan b) The Attempt at a Solution [/B] I don't know how to convert it to the form of tan (a-b) since there are...

Homework Statement The function is fA(x) = A, |x| < 1/A, and 0, |x| > 1/A Homework Equations δ(x) = ∞, x=1, and 0 otherwise The Attempt at a Solution I think the answer is the Dirac delta function, however I noticed that if you integrate fA(x) between -∞ and ∞ you get 2, which if I remember...

Homework Statement Let ##E'## be the set of all limit points of a set ##E##. Prove that ##E'## is closed. Prove that ##E## and ##\bar E = E \cup E'## have the same limit points. Do ##E## and ##E'## always have the same limit points? Homework Equations Theorem: (i) ##\bar E## is closed (ii)...

I'm having a difficult time researching the answer to my question about the speed of light. Now obviously it is a speed not only reserved for light but also all other massless particles/waves. It's obviously a constant property of our Spacetime since we can manipulate th speeds of different...

Am using Spivak and he defines limit of a function f 1. As it approaches a point a. 2.As it approaches infinity. He also defines limit f(x)=∞ x->a But though in solving exercises, we can see that all the three definitions are consistent with each other, I am not...

An interesting question has been posted by Brilliant.org. What is $\displaystyle \begin{align*} \lim_{x \to 0} \frac{x}{x!} \end{align*}$?My intuition tells me that the limit does not exist. My reasons for this are: 1. A limit can only exist if its left hand and right hand limits exist and...

The analysis of the distribution of spins for a paramagnetic solid in a B field shows that the probability of a dipole being aligned/anti-aligned with the B field ##\to 0.5## as ##T \to \infty##. The intuitive justifications that I've read say that this is "expected" as thermal motion tends to...

Homework Statement Evaluate the limit ##\lim_{x\to0} \frac{1}{x}(\frac{1}{tanx}-\frac{1}{x}) ## using Taylor's formula. (Hint: ##\frac{1}{1+c}=\frac{1-c^2+c^2}{1+c} ## may be useful) The Attempt at a Solution I began by substituting ##tanx## with ##x+\frac{x^3}{3}+x^3ε(x)##, where ε tends to...