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
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)}$$
I can see that for smaller values of $x$ the limit is 0, but what if $x$ equals infinity, wouldn't that be an indeterminate form?
The definition of a limit of a sequence,
if the limit is finite, is:
lim n >infinity un (un is a sequence) = l
<=>
∀ε> 0, ∃N: n > N => |un - l| < ε
This just means that un for n > N has to be a number for which: l -ε < un < l + ε
Now, I'm wondering, can't we just say:
n > N => |un -l| <...
Hello!
Could anybody help me?
My wondering seems so trivial, but I can't skip it.
They say that since u and d quarks are much lighter than QCD scale(~200MeV), in reality we can consider the QCD Lagrangian has an approximate global chiral symmetry with respect to these two flavors. At first, it...
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I'm trying to find the limit of this function:
$$\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)}$$
But I'm not sure of the best way to proceed.
Homework Statement
Mod note: Edited the following to fix the LaTeX[/B]
compute
##\lim_{n \rightarrow +0} \frac {8-9cos x+cos 3x} {sin^4(2x)}####\lim_{n \rightarrow +\infty} \frac {\sin(x)} x##
##\lim_{n \rightarrow +\infty} \frac {\sin(x)} x##ok find limit as x→0 for the function ##[ 8-9cos x...
We are starting sequences, and in one of the examples we have this limit:
$$\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:
$$\lim_{{x}\to{\infty}} {(-1)}^{n}{n}^{3} + {2}^{-n}$$
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...
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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.
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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.
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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...
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I have no idea how to choose a epsilon for this question.
Thanks.
When the cathode emits electrons which are accelerated towards the grid, usually on its way it will ionise a molecule in the vacuum. However at a certain low pressure there are too few molecules and therefore the electron will hit the grid and emit an x-ray. My question is, wouldn't electrons...
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...
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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...
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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)...
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2.As it approaches infinity.
He also defines limit f(x)=∞
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An interesting question has been posted by Brilliant.org.
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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...
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Hello,
I am struggling to understand a simple question on limits. I have watched a video trying to explain the theory and even have the answer right in front of me but I still don't understand. Could somebody please explain the steps in detail for me just for the first question as I'm hoping...
Homework Statement
If ##a, b \in \{1,2,3,4,5,6\}##, then number of ordered pairs of ##(a,b)## such that ##\lim_{x\to0}{\left(\dfrac{a^x + b^x}{2}\right)}^{\frac{2}{x}} = 6## is
Homework EquationsThe Attempt at a Solution
So, this is a typical exponential limit...
Homework Statement
a.) Let ##f,g:ℝ→ℝ## such that ##g(x)=sin x## and ##f(x)= \left\{
\begin{array}{ll}
x^2, x∈ℚ \\
0 , x∈ℝ\setminusℚ \\
\end{array}
\right. ##. Calculate ##\lim_{x \rightarrow 0} \frac{f(x)}{g(x)}##.
b.) Why l'Hospital rule cannot be applied here?The Attempt at a...