Limit Definition and 999 Threads

Hello mathematicians! I've recently completed a trigonometry course online and find the subject to be of great interest. I find the laws of sine and cosine fascinating and extremely useful and also, of course, Pythagoras theorem is beautiful as well. Firstly, I claim no superior knowledge...

Homework Statement Let f : (a, b) → R be a differentiable function and x0 ∈ (a, b). For any h > 0 small, there exists θ ∈ (0, 1), depending on h, such that f(x0 + h) = f(x0) + hf'(x0 + θh). If f is twice differentiable at x0 with f''(x0) != 0, prove that (a) lim h→0 (f(x0 + h) − f(x0) −...

I see that we use dimensional analysis involving constants of nature to obtain the Planck length and then apply the uncertainty principle to find the corresponding Planck mass-energy. But the energy and length scales were found by invoking a "particle" interpretation of fundamental entities of...

Homework Statement http://puu.sh/cKVxE/fb13f83a75.png In this image, I have no idea what the math behind this problem is. What exactly is happening here? Homework Equations The Attempt at a Solution I multiplied n/10 in order to get 10000n^9 as a denominator but the ln(2) confuses me...

So this goes to 0. Is this because e^(-x) going to infinity is = to 0 and thus both parts of this equation is equal to 0 ? If so, is there anyway I can prove this through my work? Thank you.

Homework Statement 2*Lim (as k approaches infinity) of (| (k/(k+1))^k |) The answer to this limit is 2/e I know there is a definition of e used, but I am unclear what to do/how to do it. If someone has a link I can look at or could point me in the right direction I would be thankful.

I'm modelling a variable output Y which has a value of 1 at x=0. I've noticed that in the system I'm modelling, as x increases, y increases at an exponentially decreasing rate, up until a limit of around 1.53. I view this as changes in x causing the Y value to increase by a max of 53%. The...

Homework Statement Given: y=\frac{\sum_{i}x_i-N\left \langle x \right \rangle}{\sqrt{N}} Show that the cumulants of y are: \begin{matrix} \left \langle y \right \rangle_c=0& & \left \langle y^2 \right \rangle_c=\left \langle x^2 \right \rangle_c & & \left \langle y^m \right...

1)The breech and barrel of a field gun have a total mass of 453.6kg. The recoil of the barrel is controlled by resistance forces from a buffer and recuperator, these forces are given by: Buffer force=71000/(s+1), recuperator force= P+ 4380s. (both are in Newtons). s is the recoil of the barrel...

What are peoples thoughts on the fact that at the exact limit of our universe. the stars are moving at just less then the speed of light away from us. If there wasn't a limit to the speed of light , isn't that what we would be seeing but the real fact then would be that it isn't the boundaries...

$\d{x}{{0}^{-}} e ^ {\frac{1}{x}}$ I am trying to solve this limit. Now, if we have $\lim{x}\to{0^{-}}1/x$ , doesn't it become $\infty$?

Hello: Prove $\displaystyle \lim_{x \to 0} \frac{x}{1 + \sin^2(x)} - 0$ Let $|x| < 1 \implies -1 < x < 1$ $\sin^2(-1) + 1 < \sin^2(x) + 1 <\sin^2(1) + 2$ $\implies \displaystyle \frac{1}{\sin^2(-1) + 1} > \frac{1}{\sin^2(x) + 1} > \frac{1}{\sin^2(1) + 1}$ $\implies \displaystyle...

Mod note: Fixed the LaTeX. The closing itex tag should be /itex, not \itex (in brackets). I find it easier to use # # in place of itex, or $ $ in place of tex (without the extra space). Homework Statement Prove \lim_{x \to 0} \frac{x}{\sin^2(x) + 1} = 0 Homework Equations Given below: The...

Homework Statement Suppose that limit x-> a f(x)= infinity and limit x-> a g(x) = c, where c is a real number. Prove each statement. (a) lim x-> a [f(x) + g(x)] = infinity (b) lim x-> a [f(x)g(x)] = infinity if c > 0 (c) lim x-> a [f(x)g(x)] = negative infinity if c < 0 Homework Equations...

Let f(x)=1+cos 2\pix and let fk=f*...*f (k-times convolution) what is the value of lim fk(1/2) when k tends to infinity Should use something about the Fourier Analysis, Could someone help me how to solve this problem?

$$\lim_{{x}\to{\pi/4}} \frac{1-\tan(x)}{\sin(x)-\cos(x)}$$ So using, L'Hospital's rule, I get: $$\lim_{{x}\to{\pi/4}} \frac{\sec^2(x)}{\cos(x)+\sin(x)}$$ But $\cos(x)+\sin(x) = 0$ when $x = \dfrac{\pi}{4}$ which is an indeterminate form, so how do I go from here?

I have this problem: $\lim_{{t}\to{0}} \frac{tan(6t)}{sin2t}$ I know sin2t = 0 when t = 0, which means the original fraction is indeterminate, so how can apply the rules for limits to solve this limit?

Good day everyone. I would like to know about the lower limit of the frequency range of active acoustic phonon modes. Textbooks say it can reach zero. But I am curious about whether it can be very small, such as 1 micro volt or the like. Would a phonon mode of so small energy be of any...

Homework Statement Show that ##\lim_{x \to a} f(x) = L## if and only if ##\lim_{x \to 0} f(x+a) = L## Homework Equations - The Attempt at a Solution For the forward direction (ie ##1 \Rightarrow 2##), I tried to first assume that 1. holds true (ie ##\forall \epsilon>0, \exists \delta>0...

Is there any known limit to the energy of a photon? I've seen a reference to pair production in the highest bracket over 1.02 MeV and I've seen references to energies from cosmic sources in the TeV range which aren't very well understood but is there any theoretical limit?

Homework Statement find lim as x,y approach 0 of (10sin(x^2 + y^2)) / (x^2 + y^2) Homework EquationsThe Attempt at a Solution direct substitution yields indeterminate form and so does multiplying by the conjugate. what other methods are there to use?

Homework Statement Is the sequence {(n!)/(n^n)} convergent or divergent. If it is convergent, find its limit. Homework Equations Usually with sequences, you just take the limit and if the limit isn't infinity, it converges... That doesn't really work here. I know I'm supposed to write out the...

I have to solve this limit. $$\lim_{{x}\to{3}} \frac{\sqrt{6x - 14} - \sqrt{x + 1}}{x -3}$$ Now, I think that by definition x - 3 is a divisor of the numerator, but how do I advance from here? Do I do long division?

My https://www.amazon.com/dp/0073532320/?tag=pfamazon01-20 gives a rule of thumb to divide by the highest power in the denominator for the following problem to demonstrate a slant (oblique) asymptote: \lim_{x\to\infty} \frac{4x^3+5}{-6x^2-7x} = \lim_{x\to\infty}...

Homework Statement I have given the statements: ##a_{n}^2 \ge x## , ##a_{n+1} \le a_{n}## , ##x > 0## and ##\inf a_{n} > 0 ##. How to prove the following: ##\lim_{n \to \infty}a_{n}=\sqrt{x}##Homework Equations ##a_{n}^2 \ge x## , ##a_{n+1} \le a_{n}## , ##x > 0## and ##\inf a_{n} > 0 ##...

Based on the following problem from http://math.uchicago.edu/~vipul/teaching-0910/151/applyingformaldefinitionoflimit.pdf: f(x) = \begin{cases} x^2 &, \text{ if }x\text{ is rational} \\ x &, \text{ if } x\text{ is irrational} \end{cases} is shown to have the following limit: \lim_{x\to 1}f(x)...

Homework Statement Okay, so the problem is to find lim (x→∞) (e^x + x) ^ (1/x) I was given the solution in the assignment in which the first step was to take the natural log of the function, then exponentiate it. i.e. lim (x→∞) [exp ln( (e^x + x) ^ (1/x))] which I understand...

1. The problem statement, all variables and given/known dat If f and g are differentiable functions with f(O) = g(0) = 0 and g'(O) not equal 0, show that lim f(x) = f'(0) x->0 g(x) g'(0) The Attempt at a Solution I know that lim as x→a f(a) = f(a) if function is continuous. since its...

Homework Statement Prove the limit \lim_{x\rightarrow 0} \frac{\log(x+1)}{x} =1 Homework Equations Use the relation 1 - \frac{1}{x} \leq \ \log x \leq x-1\ \text{if}\ x>0 The Attempt at a Solution We need to show that |\frac{\log(x+1)}{x} - 1 | \lt \epsilon\ \text{whenever}\ 0 \lt |x|...

I am giving the following input to Ti-89 Titanium: limit(S((2 x cos(100 x Pi x t)+2 x sin(10 x Pi x t))2 ,t,-t1,t1)/(2 x t1),t1,inf) where S is the integral symbol, Pi is π = 3.14, x is the multiplication symbol and inf is the infinity symbol The TI-89 answer is undef If I do it by hand it...

I am confused about the concept of a power dissipation limit for a resistor. Basically for a resistor the product of the current through it and the potential across it should not exceed 0.25 watt otherwise it starts to heat up and act in a non-linear fashion. Is this value just a constant or is...

Homework Statement "Calculate the following limit if it exists. If it does not exist, motivate why. \displaystyle\lim_{x\rightarrow 0} {\frac{x + x^2 +\sin(3x)}{tan(2x) + 3x}} Do not use l'Hôpital's rule." Homework Equations (1) \sin(a\pm b) = \cos(a)\sin(b)\pm\cos(b)\sin(a) (2)...

lim x-> -(infinity) = x + sqrt(x^2 + 2x) I know that you're supposed to multiply and divide it by it's conjugate and that the answer is -1. But I don't understand how the denominator x - sqrt(x^2 + 2x) = x + x*sqrt[1+(2/x^2)] = x[1+sqrt(1+(2/x^2)].

Hello everyone, How do I find the limit of a complex function from the definition of a limit? For instance, consider the limit ##lim_{z \rightarrow -3} (5z+4i)##. Would I simply conjecture that ##5z + 4i## approaches ##5(-3) + 4i## as ##z \rightarrow -3##; and then use the definition of a...

Homework Statement Find the ## lim _{x-> -1+} sqrt(x^2-3x)-2/|x+1| ## Homework EquationsThe Attempt at a Solution I can only solve it using l'hopital rule and would like to know the steps of solving it without using it. ## lim _{x->-1+} (2x-3)/|1|= -5/4 ##

Hello! (Wave) How can I apply L'Hôpital's rule, in order to find this limit?$$ \lim_{n \to +\infty} \frac{n^{\sqrt n}}{2^n} $$ That's what I have tried so far: $$ \lim_{n \to +\infty} \frac{n^{\sqrt n}}{2^n} =\lim_{n \to +\infty} \frac{e^{\sqrt{n} \ln n}}{e^{n \ln 2} }=\lim_{n \to +\infty}...

Homework Statement Solve the following limit: $$ \lim_{n\rightarrow \infty }n\cdot\left ( \frac{2\cdot4\cdot6 \cdots (2n-2)}{1\cdot3\cdot5\cdots (2n-1)} \right )^{2}$$ The Attempt at a Solution I don't know where to begin. Until know I've encountered limits which I could deal with in some way...

I am currently trying to learn a little about quantum mechanics, although not on very detailed level. There is one thing I wonder: What happens with the Schrödinger Equation in the classical limit, i.e. when either the mass of the particle tends to infinity or when Planck's constant tends to 0...

Here is the definition of the limit of f(x) is equal to L as x approaches a: "For every positive real number ϵ > 0 there exists a positive real number δ > 0 so that whenever 0 < |x − a| < δ, we have |f(x) − L| < ϵ." But what is the difference if I use this definition? "For every positive real...

Hi everyone. A friend of mine asked for help evaluating this multivariable limit. $\displaystyle \begin{align*} \lim_{(x,y) \to (0,0)} \frac{x\,y^8}{x^3 + y^{12}} \end{align*}$ We got the answer of 0 by converting to polars. $\displaystyle \begin{align*} \lim_{(x,y) \to (0,0)}...

Hi, Suppose you want to prove $|x - a||x + a| < \epsilon$ You know $|x - a| < (2|a| + 1)$ You need to prove $|x + a| < \frac{\epsilon}{2|a| + 1}$ So that $|x - a||x + a| < \epsilon$ Why does Michael Spivak do this: He says you have to prove --> $|x + a| < min(1, \frac{\epsilon}{2|a| +...

Compute $\displaystyle\lim_{n\to +\infty}\dfrac{1^p+2^p+3^p+\cdots +n^p}{n^{p+1}}.$

Is the following true, if no is there som theory i can studdy? ##\lim_{\Delta x \to 0}f(x+\Delta x) \cdot \Delta x = 0##

I have this problem $$\lim_{{x}\to{\infty}} \frac{2x + 1}{ \sqrt{x^2 + 2x + 1} + x}$$ How do I get from there to this step? $$\lim_{{x}\to{\infty}} \frac{2x}{ \sqrt{x^2 } + x}$$ From the last step I can calculate them limit as 1.

We had to solve this limit $$\lim_{{x}\to{\infty}} \frac{1}{x}$$ the answer is y= 0 is the Horizontal Asymptotes. I get the y = 0 but how do we know that it is the horizontal asymptote?

Ages ago I read that the maximum possible bandwidth using radio waves was in the low tbps range. I've had this rattling around in my head ever since and have occasionally tried looking it up to see if it was true. I've not had much luck in that but it did prompt me to start wondering what the...

Homework Statement Calculate the value of the limit and justify your answer with the ε-δ definition of the limit. lim (x->1) x2 Homework Equations My professor gave us the hint that we have to take δ as 0<δ≤ k0 so that δ(ε)=min{k0,ε/ (k0+2)} I'm guessing that k0 is meant to be any number...

Homework Statement find the equation of the line perpendicular to the tangent line at the given point f(x)= x√x P(1,1) Homework Equations f(a+h) - f(a) / h The Attempt at a Solution ok so first i replace (f(a) and f(a+h) in the equation x√x, and then i get 1. a+h√a+h - a√a / h, then i...

Homework Statement The lim(z->i) of [z^2+(1+i)z+2] using the epsilon-delta proof. Homework Equations z=x+iy Triangle Inequality: |z+w|<or=|z|+|w| The Attempt at a Solution For every epsilon>0, there exists a delta>0 such that |(z^2+(1+i)z+2)-(i)|<epsilon whenever 0<|z-i|<delta I'm not sure how...

People interested in quantum gravity research may wish to take note of Dittrich's September 2014 paper which I believe represents a significant step towards constructing the continuum limit and the physical Hilbert space of LQG. It will be on the third quarter MIP poll. I'll get the link...