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
I have to prove that \lim_{x \rightarrow 0^+} \left[x^\left[(\ln a)/(1+ \ln x)\right] \right]= a (in order to show that the indeterminate form of the type 0^0 can be any positive real number).
This is what I did:
Let y = \lim_{x \rightarrow 0^+} \left[x^\left[(\ln a)/(1+ \ln x)\right] \right]...
Hi, PF
This is the quote:
"If ##m## is an integer and ##n## is a positive integer, then
6. Limit of a power:
## \displaystyle\lim_{x \to{a}}{\left[f(x)\right]^{m/n}} ## whenever ##L>0## if ##n## is even, and ##L\neq{0}## if ##m<0##"
What do I understand?
-whenever ##L>0## if ##n## is even: ##m##...
I'll write my considerations which lead me to get stuck on the ##\infty-\infty## form.
$$\lim_{x \to +\infty }\sqrt{x^{2}-2x}-x+1 \rightarrow |x|\sqrt{1-0}-x+1$$
And I have no idea on how to go on...
For example: A person who gains degrees in 5 different fields. Will this person be able to refer to information learned in all 5 fields and bring it up at will?
Going by my own understanding of my self it seems to fade away when something new is learned and focused on. But once you revise over...
I have already in a previous task shown that A is not irreducible and not regular, which I think is correct. I don't know if I can use that fact here in some way. I guess one way of solving this problem could be to find all eigenvalues, eigenvectors and diagonalize but that is a lot of work and...
How could I show that this limit:
##\lim_{N\to\infty}\frac{\sum_{p=1}^N T_{4N} \left(u_0(N)\cdot \cos\frac{p\pi}{2N+1}\right)}{N}##
is equal to 0?
In the expression above ##T_{4N}## is the Chebyshev polynomials of order ##4N##, ##u_0(N)\geq 1## is a number such that ##T_{4N}(u_0)=b##, with...
Summary:: x
Let ## \{ a_{n} \} ## be a sequence.
Prove: If for all ## N \in { \bf{N} } ## there exists ## n> N ## such that ## a_{n} \leq L ## , then there exists a subsequence ## \{ a_{n_{k}} \} ## such that ##
a_{n_{k}} \leq L ##
My attempt:
Suppose that for all ## N \in {\bf{N}} ##...
A few days ago, I noticed that there is a Linux terminal command 'tc', which can be used for limiting the download and especially upload speeds of your internet connection. When I write
sudo tc qdisc add dev eth0 root tbf rate 100kbit latency 10ms burst 1540
in the terminal, the download speed...
In a statistical mechanics book, I learned about the degenerate pressure of a Fermi gas under the non-relativistic regime. By studying the low-temperature limit (T=0), we got degenerate pressure is ##\propto n^{5/3}## (n is the density).
And then I was told that in astrophysical objects, the...
If f(a) = 2, f'(a) = 1, g(a) = –1, and g'(a) = 2, the value of \lim_{x\to a}\frac{g(x)\cdot f(a)-g(a)\cdot f(x)}{x-a} is ...
A. 1
B. 3
C. 5
D. 7
E. 9
\lim_{x\to a}\frac{g(x)\cdot f(a)-g(a)\cdot f(x)}{x-a}=\lim_{x\to a}\frac{2g(x)+f(x)}{x-a}. How to determine the f(x) and g(x)? And when to use...
\lim_{x\to0}\frac{sin2x+sin6x+sin10x-sin18x}{3sinx-sin3x=}
A. 0
B. 45
C. 54
D. 192
E. 212
Either substituting or using L'Hopital gives \frac00. Is there any way to simplify it and make the result a real number?
Hello, I am currently in my college holidays and I have decided to do some maths to improve. My weakness is graphing and I am hoping to get some help or the solution on this question.
Question:
Let P(k,k^2) be a point on the parabola y=x^2 with k>0.
Let O denote the origin.
Let A(0, a)denote...
I know it's probably an easy one, but I'm getting confused on how to treat that exponential numerator in order to escape from the indeterminate form ##\frac{0}{0}##
I'm trying to determine why
$$ \lim_{N \rightarrow +\infty} ln( \frac {N!} {(N-n)! N^n}) = 0$$
N and n are both positive integers, and n is smaller than N. I want to use Stirling's, which becomes exact as N->inf:
$$ ln(N!) \approx Nln(N)-N $$
And take it term by term:
$$ \lim_{N...
Given is the following: lim x-2 of f(x)=2 prove (using delta, epsilon definition of a limit) that a delta exists so that when [x]<delta then f(x)>1
I came up with when [x-a]<delta (f(a)-epsilon<f(x)< f(a) + epsilon) so f(a)-epsilon>1 so epsilon<f(a) -1 but I don't know how to prove this or...
(a) I thought perhaps a parameterization would be the place to begin given all the squared terms.
x=rcos(u)sin(v)
y=rsin(u)sin(v)
z=rcos(v)
That would yield: r^k(cos(v))^k*(e^(r^2*(sin(v))^2))/(r^(2k))
Canceling a r^k at each level: (cos(v))^k*(e^(r^2*(sin(v))^2))/(r^(k))
I'm not sure how...
Attached is a proof that $$ lim \inf \subset lim \sup $$ for an infinite sequence of non-empty sets. The basic idea is to use the axiom of choice/well ordering theorem to show that 1) there is something in the limit superior 2) there is something in the limit superior that is not in the limit...
If a voltage source is sinusoidal, then we can introduce a phasor map and come up with equations like$$V_0 e^{i \omega t} = I(R + i\omega L + \frac{1}{\omega C} i)$$where ##I## would also differ from ##V## by a complex phase.
But if you set ##\omega = 0##, which would appear to correspond to...
The apparent length of a rod is determined to be$$\tilde{L}(x_0) = \gamma L + \beta \gamma \sqrt{D^2 + (\gamma x_0 - \frac{L}{2})^2} - \beta \gamma \sqrt{D^2 + (\gamma x_0 + \frac{L}{2})^2}$$I am trying to determine expressions for ##\tilde{L}(x_0)## when ##x_0 \rightarrow -\infty## and ##x_0...
The Schwarzschild metric seems to model, for example, the earth’s gravity field above the earth’s surface pretty well, even though the Earth is not really a golf-ball sized black hole down at the center. Can the same be said for the Kerr metric? Does it model a rotating extended body’s gravity...
All the references I find refer to safely charging lithium cells by a method like this:
https://www.powerstream.com/li.htm
The next page shows the effects on capacity of charging to less than the 4.2 V terminal cell voltage. For example, charging to 4.0 V still provides 73% of the capacity...
Hello everyone, I need to find this limit . What I tried is that
,
but clearly, 1/x diverges so I don't think it was very helpful.
Could someone help me what I need to do please?
I need to solve this limit without L'Hôpital's rule. Could someone give me a hint what
I need to do please? I just can't find this algebraic trick. Thank you in advance!
We have the limit of the sequence ##\frac{a^n}{n}## where ##a>1##. I know it is ##+\infty## and i can prove it by switching to the function ##\frac{a^x}{x}## and using L'Hopital.
But how do i prove it using more basic calculus, without the knowledge of functions and derivatives and L'Hopital...
(1 + a^x)^(a^(-x))
Let's assume a, say, two.
the limit of it, with x tending to infinity, is one, but i was thinking...
Calling 2^x by a, we have that when x tend to infinity, so do a, So:
that is euler number no? Contradictory... where am i wrong?
Hello everybody, could you help me with this problem please? I have to find a derivative in x0 of this function (without using L'Hospital's rule):
I used the definition , but I don't know what to do next. Thank you.
If there is no ##(-1)^2## factor, I can find the limit. But, now I have no idea how to find limit for the ##(-1)^\infty##. I thought ##(-1)^\infty## is an indeterminate form. So, how to modify this? Thanks!
I tried to substitution n = infinity so I got (infinity)*sin (1/infinity). I thought 1/infinity is approaching zero. So, sin (1/infinity) is same with sin (0). With these idea, my solution is lim n--> infinity n sin(1/n) = 0.
But, the answer book say that the answer is 1.
I tried another...
Hello,This is actually a piece of a little bigger problem (convergence of a series) - you can see the ratio test ak+1 / ak
That's why the (n) and (n+1) terms
I have lim n->∞ of (n√n) / (n+1)√(n+1) ∞/∞
I have tried L'Hopitals rule (requiring multiple times) and I am not seeing an end...
I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 1, Section 1.2: Topological Spaces ...
I need help in order to fully understand Singh's proof of Theorem 1.3.7 ... (using only the definitions and results Singh has established to...
(NOTE: I have had a few similar postings lately on this subject, but they were much broader in scope, so I am posting only for this particular case; everything else has been figured out.)
If given that
limx -> a f( x ) = +∞
limx -> a g( x ) = +∞
what is the epsilon-delta formulation for...
What is exactly the definition of symmetric limit?
It's the first place in the book that I see this notation, and they don't even define what it means.
How does it a differ from a simple limit or asymptotic limit?
I found a few hits in google, but it doesn't seem to help...
I’ve read many Legends and Canon Star Wars books and I always take away stuff on their limits of technology and science. Over the years; here are some things they said science can’t do.
1.) Cybernetic liver- In Lost Stars, it was said Ciena’s liver could not be replaced as it was one of the...
I have the function: ##\sqrt{\left(\frac{x}{h}+1\right)^{2}+\left(\frac{y}{h}\right)^{2}}-\sqrt{\left(\frac{x}{h}\right)^{2}+\left(\frac{y}{h}\right)^{2}}##
I would like to find an analytical solution, the equivalent function, in the limit of h approaching zero.Additional info which might be...
A 2019 paper by Gorecki et al. derives an uncertainty principle limit that is larger than the conventional Heisenberg limit by a factor of ##\pi##:
https://arxiv.org/abs/1907.05428
I'm wondering if any QM experts have seen this and what your thoughts are.
Hello all,
Given following limits:
##\lim_{x \rightarrow 1} {\frac {\sqrt x -1} {x^2 - 1}}##
##\lim_{x \rightarrow 1} {\frac {\sqrt {x+1} - 2} {x - 3}}##
##\lim_{x \rightarrow 1} {\frac {\sqrt[3] x - \sqrt[4] x} {\sqrt[6] x - \sqrt x}}##
Those limits can be evaluated by letting ##x = t^2##...
Someone asked me what a dime was (this is UK.) I, not knowing, nevertheless promptly replied, it must be 5 cents, because they called their 55mph speed limit the double dime. Then of course went to Google to check and found that it is 10 cents.
How does double 10 become 55? So back to Google...
Hey! :o
I want to check the existence of the limit $\lim_{x\to 0}\frac{x}{x} $ using the definition.
For that do we use the epsilon delta definition?
If yes, I have done the following:
Let $\epsilon>0$. We want to show that there is a $\delta>0$ s.t. if $0<|x-0|<\delta$ then...