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
Homework Statement
lim as z--> i , \frac{z^2-1}{z^2+1}
The Attempt at a Solution
[/B]When we plug in i we get -2/0, so we get division by 0, Does this mean the limit is
infinity, I also tried approaching from z=x+i where x went to 0, you get the same answer,
I also approached from...
Homework Statement
limit (x -> 0 y -> 0) of xy/sin(x+y)
Homework Equations
None that come to mind but maybe Lopital's Rule
The Attempt at a Solution
I know that the limit does not exist but I am having trouble figuring out how to show that it does not
using the line x=y gives x^2/sin(2x)...
In your opinion, how old is too old to consider a four-year math degree? Can a person who is middle age return to college to major in math even if the degree itself will not lead to a rewarding career?
I can't prove it and I've got it by some intuition because not many properties of superlogarithms are known. I don't think anyone can prove it but is there some way to at least check if it is correct.
The limit is:
$$\lim_{h\rightarrow0}slog_{[log_xx+h]}[log_{f(x)}f(x+h)]$$
where ##slog## is the...
Homework Statement
##\frac{e^x-1}{x}##
Evaluate the limit of the expression as x approaches 0.
Homework Equations
3. The Attempt at a Solution [/B]
The question i have is more theoretical. I was able to solve this problem by expanding the expression into the talyor polynomial at ##x=0##. I...
Hello all,
I have a complicated function:
\[f(x)=\left ( e^{x}+x \right )^{^{\frac{1}{x}}}\]
I need to find it's derivative and it's limit when x goes to infinity.
As for the derivative, I thought maybe to use LN, so that I can get rid of the exponent, am I correct?
How should I approach...
Homework Statement
Hi
I am stuck on a small algebra set in the weak limit theorem to recover Newtonian equations
The text I am looking at:
##\frac{d^2x^i}{ds^2}+\Gamma^i_{tt}\frac{dt}{ds}\frac{dt}{ds}=0## (1)
##\Gamma^{i}_{tt}=-1/2 \eta^{ij}\partial_{j}h_{tt} ## (to first oder in the...
Homework Statement
Suppose a_{n}=\frac{n^2-2n+1}{2n^2+4n-1}
For each positive number \epsilon , find a number N such that:
\mid a_{n} - L\mid < \epsilon whenever n > N.
Homework EquationsThe Attempt at a Solution
\mid \frac{n^2 -2n + 1} {2n^2+4n-1} - \frac{1} {2} \mid < \epsilon...
So, a black hole has infinite gravity that even light can't escape from it,
my question is,
the gravitational field of a black hole can even pull light into it,
then it means it is even faster than light, if not, light can escape from it.
Does this argument make any sense, please tell me! thanks
I am given an implicit expression for an algebraic function, ##w(z)## as:##f(z,w)=(1-3 z+3 z^2-z^3)+(-4+8 z-4 z^2)w+(6-6 z)w^2+(-4)w^3+(1)w^4=0## with ##\displaystyle \frac{dw}{dz}=-\frac{\frac{df}{dz}}{\frac{df}{dw}}=\frac{6 w^2+8 w z-8 w+3 z^2-6 z+3}{4 w^3-12 w^2-12 w z+12 w-4 z^2+8 z-4}##...
It is said that the physical vacuum, is by definition a state with no "physical particles" -- more precisely, it is the ground state (state of lowest energy) of the field.
Is there any beyond standard model where the vacuum is a lower energy limit of another theory or stuff... like the...
Homework Statement
For which value of d does the following limit exist?
lim x->d ln [ (x2-13x+30) / (x-d) ]
Homework Equations
None
The Attempt at a Solution
I understand how to find limits when the limit goes to a real number, and has a variable in the function to solve for, but not when...
Hello
I am trying to solve this limit here:
\[\lim_{x\rightarrow -\infty }\sqrt{x^{2}+3}+x\]
I understand that it should be 0 since the power and square root cancel each other, while the power turned the minus into plus, and then when I add infinity I get 0. This is logic, I wish to know how...
In this solution, in the last 3rd line, I get the first part (-e^-1 - e^-1), however, after the '-' symbol, the person writes (1/b * e^1/b - e^1/b) and takes the limit as b->0. However, shouldn't this give him (inf. * e^inf - e^inf)?
Thanks
(His answer is correct, by the way)
We kept getting some voltage spikes back into our amplifier. One of the things we put in place is a 10kOhm resistor (discharge load) connected to the output of the amplifier. However, the resistor keeps burning and setting on fire.
The amp company offered a way on checking this...
I'm really...
Homework Statement
I would like to understand how the limit was changed in the ratio test from step 1 to step 2 in the image that I've posted. I thought that the denominator would look like (2/n+2)(2/n+1) in step 2 since it looks like we are just turning the n's into reciprocals. Any help here...
Homework Statement
Using the taylor series at point ##(x=0)## also known as the meclaurin series find the limit of the expression:
$$L=\lim_{x \rightarrow 0} \frac{1}{x}\left(\frac{1}{x}-\frac{cosx}{sinx}\right)$$
Homework Equations
3. The Attempt at a Solution [/B]
##L=\lim_{x \rightarrow 0}...
Homework Statement
$$\lim_{x\to\infty} \left(\frac{n^2+2n+1}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}$$
Homework Equations
3. The Attempt at a Solution [/B]
I tried
##\lim_{x\to\infty} \left(\frac{n^2+2n+3-2}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}=##
##\lim_{x\to\infty}...
Homework Statement
Prove that
$$ \lim_{x\to 0} \sqrt{x^3+x^2}\; \sin\left(\frac{\pi}{x}\right) = 0 $$
using Sandwich theorem
Homework Equations
Sandwich Theorem
The Attempt at a Solution
Now we know that sine function takes values between -1 and 1. ## -1 \leqslant...
Some claim not possible naturally, ie beat 2 hours. While not possible under race conditions Nike is trying to see if it is humanly possible;
http://running.competitor.com/2016/12/news/nike-launches-program-break-2-hour-marathon-barrier-2017_160032
Basically i need to evaluate the limit of this expression ##\lim \sqrt[3]{n^6-n^4+5}-n^2=?## I want to know if this is correct and why:
##\lim \sqrt[3]{n^6-n^4+5}-n^2=\lim...
Hello all. I am working on proving some theorems about Monte Carlo simulation and have proven a theorem that, in a certain formula, it is valid to replace a random variable in the denominator of a fraction by its expected value. I have been wondering whether this result can be generalised to...
Hey! :o
I want to calculate the limit $$\lim_{x\rightarrow \infty}x^{100}\left [\frac{1}{x}\right ]$$
When $x\rightarrow +\infty$ it holds that $0<\frac{1}{x}<1$, or not? (Wondering)
If yes, it holds that $\left [\frac{1}{x}\right ]=0$ or not? Then $x^{100}\left [\frac{1}{x}\right ]=0$, and...
Homework Statement
I was sick and missed the lecture so having hard time with this problem @_@.
http://i.imgur.com/ByK7iVk.png
Homework Equations
I don't know how to solve it all the textbook don't have particular problem so having a hard time figuring it out.
The Attempt at a Solution
My...
I have
$\lim_{{x}\to{0}} lnx \cdot x$
$x$ approaches 0, and $lnx$ approaches $\infty$.
How can I reason about this.
I suppose $x$ approaches 0 more quickly than $lnx$ approaches $\infty$ , therefore it is zero. Is this accurate? How can I prove this.
Here's a graph and its triple integral. How are the limits of integration for the outer integral [-2,2]? I have no idea how this was found.
Any help would be appreciated!
Mods, I wasn't sure whether to put this in quantum physics or relativity, but since the speed of light is the limiting factor I chose here. Move wherever you think is best.
Okay so the speed of light is the asymptote for the speed that objects can accelerate to, and the Planck time is the...
In Schutz says When we have weak gravitaional fields then the line element *ds* is
$$
ds^{2}=-(1+2\phi)dt^{2}+(1-2\phi)(dx^{2}+dy^{2}+dz^{2})
$$
so the metric is
$$
{g_{\alpha\beta}} =\eta_{\alpha\beta}+h_{\alpha\beta}= \left( \begin{array}{cccc}
-(1+2\phi) & 0 & 0 & 0\\
0 & (1-2\phi) & 0 &...
Homework Statement
http://prntscr.com/dcfe0u
Homework EquationsThe Attempt at a Solution
So I'm not really strong in proofs but I think you may be able to do something like this:$$lnL = \frac{ln(1+1/x)}{x}$$
$$lnL = \frac{1/x^2}{1+1/x}$$
and then more simplifying I get something like:
$$lnL =...
One way that people introduce the Hawking temperature of an event horizon, is by taking the near-horizon limit of the BH metric and then do a Wick rotation of the time coordinate. Then, the regularity of the metric requires that the Euclidean time to be periodic. But how can this give us the...
Hey! :o
I want to find the $\lim\sup$, $\lim\inf$ and the limit points of the following sequences:
1. $a_n=(-1)^n\frac{3n+4}{n+1}$
2. $a_n=\sqrt[n]{n+(-1)^nn}$
3. $a_n=\left ( \frac{n+(-1)^n}{n}\right )^n$
4. $a_n=(-1)^{\frac{n(n+1)}{2}}\sqrt[n]{1+\frac{1}{n}}$ I have done the following...
Hey! :o
Let $(a_n)_{n=1}^{\infty}$ be a real sequence and let $(b_n)_{n=1}^{\infty}$ a sequence in the set of limit points of $(a_n)_{n=1}^{\infty}$, $L(a_n)$.
There is also a $b_0\in \mathbb{R}$ with $b_n\rightarrow b_0$ for $n\rightarrow \infty$.
I want to show that then $b_0\in L(a_n)$...
How could I calculate:
$\displaystyle \lim_{n \rightarrow + \infty}{(\sqrt(n^2+n) - \sqrt[3](n^3+n^2))}$
Everything that I tried I always got infinite forms or 0 in denominator. I don't have any idea what else should I try here.
I've got another limit question here, but I don't suppose you use the same method as last time for sure (rational).
I essentially got (infinity)^0 and just assumed that to be equal to 1 which made sense to me. However, in my textbook and notes, (infinity) to the power of 0 is one type of an...
Hi, I am having trouble with these kind of questions where we have to use L'Hospital's Rule. I took the ln of the function to get the x out of the exponent, and then followed the Rule by taking the derivative of the top and bottom (using a shortcut we learned: lim x --> infty f(x)g(x) = lim x...
Homework Statement
can you cancel the x and the |x|[/B]
lim x→0 ( x(1-(cos(x))/|x| )
2. Homework Equations The Attempt at a Solution
At first i thought the limit would just be undefined as x approaches 0 but the answer to the problem is actually 0, so can you just cancel the x with the...
Homework Statement
Find the limes of the equation, so lim n -> ∞
Homework Equations
[/B]
The Attempt at a Solution
I tried to solve this by using the third binomial formula and formed this to
I wanted to show that it's <= 1/n, but then I checked wolfram alpha and it seems like it actually...
Homework Statement
[/B]Homework Equations
delta(x) = [b-a]/n
xi=a+delta(x)i
The Attempt at a Solution
So, it is said that i have to use riemann sums to solve this one.
what i did is i took the 1/12k out thus getting
1/12k[1/[1+1/12k] + 1/[1+2/12k] + 1/{1+19k/12k]]
I found that
xi =...
Homework Statement
$$\lim_{x \to -∞}{\sqrt{x^2 + bx + c} - x}.$$Homework EquationsThe Attempt at a Solution
So in problem 1, once I got to a point where I am to divide by the highest power in the denominator(x) I get something like:
$$\lim_{x \to -∞}\frac{bx+c}{\sqrt{x^2+bx+c}+x}$$
Now what I...
Q's Let f,g ℝ→ℝ. Suppose that g is bounded. This means that its image is bounded or in other words there exists a positive real number B s.t. |g(x)| ≤ B ∀ x. Prove that if lim x→c f(x) = 0, then lim x→c f(x)g(x) = 0.
Work.
See the picture.
I am really confused I can't seem to understand the idea...
Hello! I have a question that has been bothering me since I first started learning about Special Relativity:
Given only the Minskowskian metric and/OR the spacetime interval, how can one reach the conclusion that the speed of light is invariant for every observer and how can one conclude that it...
This question comes from the equation E = vB (moving conductor in a magnetic field -- E = electric intensity, v= speed of the conductor's movement, and B = magnetic field strength). Say B is constant, so the only thing we have to rely on to vary the electric intensity in the conductor is its...
Dear PF Forum,
What is the limit of acceleration?
I've been reading old threads, and I found this.
G = 6.673 x 10-11 N m2/kg2
Solar mass = 1.989 * 1030kg
And I tried to plug in some numbers...
In a distance 30 km from a 10 solar mass object the acceleration is...
##a =...
What does it mean for a ##f(x,y)## to be differentiable at ##(a,b)##? Do I have to somehow show ##f(x,y)-f(a,b)-\nabla f(a,b)\cdot \left( x-a,y-b \right) =0 ##? To show the function is not though, it's enough to show, using the limit definition, that the partial derivative approaching in one...