In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (i.e., eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.
Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.
The limit inferior of a sequence
x
n
{\displaystyle x_{n}}
is denoted by
lim inf
n
→
∞
x
n
or
lim
_
n
→
∞
x
n
.
{\displaystyle \liminf _{n\to \infty }x_{n}\quad {\text{or}}\quad \varliminf _{n\to \infty }x_{n}.}
The limit superior of a sequence
When does an on/off ratio become impractical? I've read that typical CMOS devices have an on/off ratio of 10^6-10^10. Is there a lower limit that prevents logic devices from functioning appropriately? As in, what is a practical lower limit for the on/off ratio but still enable logic functionality?
Would this ##f'(x)>g'(x)\,\forall x\in [a,\infty)\text{ and }f,\,g\underset{\infty}{\to}0\Rightarrow \lim_{x\to\infty}f(x)/g(x)=0## hold if ##\frac{f(x)}{g(x)}\neq c##?
Hello all,
I am trying to solve a limit:
\[\lim_{x\rightarrow 0}\frac{sinh (x)}{x}\]
I found many suggestions online, from complex numbers to Taylor approximations.
Finally I found a reasonable solution, but one move there doesn't make sense to me.
I am attaching a picture:
I have marked...
Hello everyone,
I want to calculate the following limits:
\[\lim_{x\rightarrow \infty }\frac{[x\cdot a]}{x}\]
using the sandwich rule, where [xa] is the integer part function defined here:
Integer Part -- from Wolfram MathWorld
I am not sure how to approach this. Any assistance will be...
I'm not sure if the way I solve these limits is correct, so let me know if I'm doing something wrong.
$$\lim_{x^2+y^2 \rightarrow +\infty} {\frac {xy} {x^2+y^2}}$$
$$r = x^2+y^2$$
$$\lim_{r \rightarrow +\infty} {\frac {r\cdot cos(v) \cdot r \cdot sin(v)} r}$$
$$\lim_{r \rightarrow +\infty}...
I am currently starting with my first qft lectures and i am trying to see for the free particle that the propagator $$ <x_i | e^{-i\frac{p}{2m} T|x_f}>$$ will equal to one if x_f = 1, x_i=0 m=1 u=1 p=1, T=1 and $$\hbar \rightarrow 0$$ or 0 otherwise. I understand that this limit will result in...
As I have studied before, I found that Infinite Red Shift occurs where gtt = 0 but this exercise says that on Kerr's Black Hole it doesn't really work like that.
Right now I'm blocked because I didn't find anything on the internet about it so I don't know how to show this phenomenon. Any help...
I was reading that in inverse scattering approach, we divide the region of interest into discrete grids and size of each grid should be much smaller than the incident wavelength (usually smaller than one-tenth of wavelength).
By this logic, theoretically, I can use inverse electromagnetic...
I don't know what do do from here other than i can make the 3/e^x a 0 due to the fact its divided by such a large number. What do i do with the e^-3x? Thanks for the help
We have so many great books available for Calculus, such as : Spivak's Calculus, Stewart Calculus, Thomas Calculus , Gilbert Strang's Calculus, Apostol's Calculus etc.
These books are very nice but they teach you the concepts well and all the standard techniques that are available for solving...
The Propane industry mandates that a tank not be filled more than 80%. The question I have is this: how do I calculate the limit of liquid propane in a standard 3800 liter tank given a 30 degree rise in temperature (from 273 K to 303 K) such that it will not rupture the tank? For example, can I...
What does the reconstructed wave look like if we sample the input an infinitesimal amount under the Nyquist limit? I can intuitively picture how we can (ideally) reconstruct an input sampled at the Nyquist limit (and appropriate phase) because we are able able to get the extreme values of the...
*Given: δ = |all real numbers|
*Given: ϵ = |all real numbers|
For any x value within +/- |δ| of c, we can find a y=f(x) within the corresponding +/-|ϵ| of L. According to my professor, the mathematical representation of this is |x-c| < δ and |f(x) - L | < ϵI fail to understand why it cannot...
Find the limit as x approaches 0 of x2/(sin2x(9x))
I thought I could break it up into:
limit as x approaches 0 ((x)(x))/((sinx)(sinx)(9x)).
So that I could get:
limx→0x/sinx ⋅ limx→0x/sinx ⋅ limx→01/9x.
I would then get 1 ⋅ 1 ⋅ 1/0. Meaning it would not exist.
However the solution is 1/81...
I'm getting 2119.36 for the concentration of mg/cubic meter of this substance...it just feels wrong though.
Steps I followed:
First, I figured out how many grams of the substance there were using the density formula, then saw how many were present per cubic meter after calculating the volume...
If one approaches the origin from where ##x_2=0##, the terms ##x^2_1x_2+x^2_2x_3## in the denominator equal ##0##. Substituting ##|\textbf{x}|^2## for ##t## yields the expression ##\frac{e^t-1}{t}##, which has limit 1 as ##\textbf{x}\to\textbf{0}## and thus ##t\to0##. So the limit should be 1 if...
Since $$\lim_{x \rightarrow 0} \frac {R_{n,0,f}(x)} {x^n}=0,$$ ##P_{n,0,g}(x)## contains only terms of degree ##\geq 1## and ##R_{n,0,g}## approaches ##0## as quickly as ##x^n##, I can most likely prove this using ##\epsilon - \delta## arguments, but that seems overly complicated. I also can't...
Consider limx→3x^2=9.
Find a maximum value of δ such that:
|x2 - 9|<0.009 if |x-3|<δ
I just learned how to do this today and I am quite comfortable doing this if the function is linear, however now I am struggling with working with quadratic functions.
So far this is what I have come up with...
I don't know how to show that this limit is zero.
It seems that ##\sum_{i=1}^N a_{i,N} /N = 1## and the fact that ## 0 < a_{i,N} < M > 1## implies that some ##a_{i,N}## are less than one.
Another conclusion I guess is correct to draw is that ##\lim_{N \to \infty} \sum_{i=1}^N a_{i,N}^2 /N < 1##.
Instinct tells me to just plug in the number, say the limit is zero, and be done with it. But at the same time, while reading the statement from the "Relevant equations" section of this post, I cannot feel but feel some doubt as to whether or not this is the right approach. I mean, only the...
Let's say you are rubbing a balloon on your hair to make it charged. If you then discharge the balloon and rub it on your hair again (and repeat this process numerous times). Would your hair run out of electrons so eventually you would be unable to charge the balloon, or would your hair gain...
This question consists of two parts: preliminary and the main question. Reading only the main question may be enough to get my point, but if you want details please have a look at the preliminary.
PRELIMINARY:
Let potential due to a small volume ##\delta## at a point ##(1,2,3)## inside it be...
The classical limit of QM that have always puzzled me. There are common statement saying that you can recover classical mechanics by taking the limit of h->0 or by taking large quantum numbers. Other times times the Erhenfest theorem or the Madelung/hydrodynamics version of the Schroringer...
Let A ⊆ R, let f : A → R, and suppose that (a,∞) ⊆ A for some a ∈ R. Then the
following statements are equivalent:
i) limx→∞ f(x) = L
ii) For every sequence (xn) in A ∩ (a,∞) such that lim(xn) = ∞, the sequence (f(xn))
converges to L.
Not even sure how to begin this one, other than the fact...
I simplified somewhat and got (1/e-(1-x/(1+x))x)/(1/x)
So i can't find that it is 0/0 form so tried by applyying L'Hospitale,But it just became complicated.So need help.
I want to show that the limit of the following exists or does not exist:
When going along the path x=0 the limit will tend to 0 thus if the limit exists it will be approaching the value 0
when going along the path y=0, we get an equation with divisibility by zero. Since this is not possible...
Let's say I have a function whose derivative is (tan(x)-sin(x))/x. It is not defined for X=0 but as X approaches 0 the derivative approaches 0, so should I conclude that my function is not differentiable at X=0 or should I conclude that the derivative at X=0 is 0.
...Or even based on logic?
I understand that it is expected that there might be a Universal Speed Limit and that this seems with extremely high probability to coincide with the speed of em transmission in a vacuum.
This is borne out by experimentation and observation.
Are there any other...
I wrote cos(pi(n^2+n)^(1/2)) as cot(pi(n^2+n)^(1/2))/cosec(pi(n^2+n)^(1/2)) and as we know cot(npi)=infinity and cosec(npi)=infinity , so i applied L'Hospital.After i differentiated i again got the same form but this time cosec/cot which is again infinity/infinity.But if i differentiate it i...
Hey! :o
We consider the equation \begin{equation*}u_{k+1}=\begin{pmatrix}a & b \\ 1-a & 1-b\end{pmatrix}u_k \ \text{ with } \ u_0=\begin{pmatrix}1 \\1 \end{pmatrix}\end{equation*}
For which values of $a$ and $b$ is the above equation a Markov process?
Calculate $u_k$ as a function of $a,b$...
I've recently been reading about the 2-dimensional Ising model and its continuum limit from several sources, including
https://webhome.weizmann.ac.il/home/fnfal/papers/Ising/lecture1.pdf
https://webhome.weizmann.ac.il/home/fnfal/papers/Ising/lecture2.pdf
As far as I understood it, the state...
Is it known how much momentum a photon possesses if it's wavelength is at the Planck length, and what happens if it's momentum is somehow increased from that?
Let:
##\displaystyle f=\int_{V'} \dfrac{x-x'}{|\mathbf{r}-\mathbf{r'}|^3}\ dV'##
where ##V'## is a finite volume in space
##\mathbf{r}=(x,y,z)## are coordinates of all space
##\mathbf{r'}=(x',y',z')## are coordinates of ##V'##
##|\mathbf{r}-\mathbf{r'}|=[(x-x')^2+(y-y')^2+(z-z')^2]^{1/2}##...
I am working with a simulation which generates an arbitrary number ##n## of identical curves with different phases and calculates their (normalized) sum. As expected, the fluctuation depth of the curves decreases as we increase ##n##. Here is an example of my simulation (when ##n>1##, the...
Express the limit ##lim_{n\rightarrow\infty} \sum_{i=1}^n \frac2n\ (1+\frac {2i-1}{n})^\frac13##
This is worked example but I would like to ask about the points I don't understand in the book.
"We want to intepret the sum as a Riemann sum for ##f(x)=(1+x)^3## The factor ##\frac2n## suggests...
Let $m_n$ be the smallest value of the function:
$$f_n(x)=\sum_{k=0}^{2n}x^k.$$
Show, that $m_n\to\frac{1}{2}$ as $n \to \infty$.
Source: Nordic Math. Contest