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
Given a singular matrix ##A##, let ##B = A - tI## for small positive ##t## such that ##B## is non-singular. Prove that:
$$
\lim_{t\to 0} (\chi_A(B) + \det(B)I)B^{-1} = 0
$$
where ##\chi_A## is the characteristic polynomial of ##A##. Note that ##\lim_{t\to 0} \chi_A(B) = \chi_A(A) = 0## by...
I have the following definition:
$$ \lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$
From this, how can I get the definition of
$$\lim_{x\to p^-}=-\infty? $$
Problem: If sequence ## (a_n) ## has ##10-10## as partial limits and in addition ##\forall n \in \mathbb{N}.|a_{n+1} − a_{n} |≤ \frac{1}{n} ##, then 0 is a partial limit of ## (a_n) ##.
Proof : Suppose that ## 0 ## isn't a partial limit of ## (a_n) ##. Then there exists ## \epsilon_0 > 0 ## and...
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}} ##...
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...
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...
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##...
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...
<Moderator's note: Moved from a technical forum and thus no template.>
$$\lim_{x\rightarrow 0} (x-tanx)/x^3$$
I solve it like this,
$$\lim_{x\rightarrow 0}1/x^2 - tanx/x^3=\lim_{x\rightarrow 0}1/x^2 - tanx/x*1/x^2$$
Now using the property $$\lim_{x\rightarrow 0}tanx/x=1$$,we have ...
Homework Statement
lim (1/x - 1/3) / (x-3)
x->3
Homework Equations
The Attempt at a Solution
I tried to cancel the bottom (x-3) out by multiplying the top by 3/3 and x/x and then got ((3-x)/3x)/(x-3) but ended with 0/0 and the right answer is -1/9. The top part is confusing me.
<Moderator's note: Moved from a technical forum and thus no template.>
$$\lim_{x \to 0} \cos(\pi/2\cos(x))/x^2$$
I tried to evaluate the limit this way,
$$\lim_{x \to 0} \cos(\pi/2\cdot1)/x^2$$ since $$\cos0=1$$
$$\lim_{x \to 0} \cos(\pi/2\cdot1)/x^2=\lim_{x \to 0} 0/x^2$$
Now apply...
I've understood the formal definition of limits and its various applications. However, I'm trying to dive more into the history of how the concept of limits were conceived (more than what Wikipedia tends to cover), and how to formally understand and visualise infinitesimals.
For example, I know...
Homework Statement
Consider the standard square well potential
$$V(x) =
\begin{cases}
-V_0 & |x| \leq a \\
0 & |x| > a
\end{cases}
$$
With ##V_0 > 0##, and the wavefunctions for an even state
$$\psi(x) =
\begin{cases}
\frac{1}{\sqrt{a}}cos(kx) & |x| \leq a \\...
Homework Statement lim x~∞ 〈√(x⁴+ax³+3x²+ bx+ 2) - √(x⁴+ 2x³- cx²+ 3x- d) 〉=4 then find a, b, c and d[/B]
Homework Equations
all the methods to find limits
The Attempt at a Solution
it can be said that the limit is of the form ∞-∞.I am completely stuck at this question.the answer is a=2...
Homework Statement lim x~a 〈√(a⁺2x) -√(3x)〉 ÷ 〈√(3a+x) - 2√x〉[/B]
Homework Equations rationalisation and factorisation[/B]
The Attempt at a Solution i had done rationalisation but the form is not simplifying.plzz help me.[/B]
Homework Statement
a. Compute the limit for f(x) as b goes to 0
Homework Equations
$$f(x) = \frac{(a+bx)^{1-1/b}}{b-1}$$
##a \in R##, ##b\in R##, ##x\in R##
The Attempt at a Solution
##a+bx## goes to ##a##
##1/b## goes to ##\infty## so ##1-1/b## goes to ##-\infty##
##(a+bx)^{1-1/b}## then goes...
When we define a limit of a function at point c, we talk about an open interval. The question is, can it occur that function has a limit on a certain interval, but it's extension does not? To me it seems obvious that an extension will have the same limit at c, since there is already infinitely...
Homework Statement
[/B]
We have a function f(x) = |cos(x)|.
It's written that it is piecewise continuous in its domain.
I see that it's not "smooth" function, but why it is not continuous function - from the definition is should be..
Homework Equations
[/B]
We say that a function f is...
Homework Statement
Can I use L'Hopital's rule here. What I get as a solution is -30/-27 while in the notebook,
without using the L'Hopital's rule the answer is -(2/27)
The attempt at a solution
The derivatives i get are:
x/(x2+5)½
(3x2+2x)/3(x3+x2+15)⅓
2x-5
½ and ⅓ are there because it's...
Homework Statement
##\lim_{h \to 0} \frac{f(x - 2h) - f(x + h)}{g(x + 3h) - g(x-h)}##
While f(x) = cos x
g(x) = sin x
Homework Equations
The Attempt at a Solution
Using L Hopital i couldn't make it more simple.
I tried to divide it by cos and sin
Can you give me clue?
Homework Statement :
Find [/B]
limx->∞(xα(sin2x!)/(x+1)
α∈(0,1)
Options are:
a)0
b)1
c)inifinity
d)does not exist
Homework Equations : -[/B]
The Attempt at a Solution :
limx->∞(xαsin2x!)/(x+1)[/B]
Dividing the numerator and denominator by xα,
we have:limx->∞sin2x!)/(x1-α+x-α)
clearly x−αis...
Homework Statement :[/B]
limx->0xsin(1/x)
Homework Equations : [/B]-
The Attempt at a Solution :[/B]
I feel the limit does not exist. Because sin(1/x) is largely changing value as x approaches 0,(since it is an oscillating function), and in limit, we check what happens in neighborhood of the...
Homework Statement
Finding the value of the limit:
$$\lim_{t\to +\infty} t+\frac{1-\sqrt{1+a^2t^2}}{a}$$
##a## is just a costant
The Attempt at a Solution
At first sight I had thought that the limit was ##\infty## but then I realized that there is an indeterminate form ##\infty - \infty##. I...
I tried to find the integral of x^m using the definition of Riemann summation. Everything went smoothly until the limit of ∑n=1kn^m divided by k^( m+1), when k approached infinity, showed up.
It is clear that it approaches to 1/m+1, but it has to be proved, of course.
One could induce that fact...
Hello. I have problem with this integral :
\lim_{n \to \infty } \int_{\mathbb{R}^+} \left( 1+ \frac{x}{n} \right) \sin ^n \left( x \right) d\mu_1 where ## \mu_1## is Lebesgue measure.
Hello, I would like to begin by saying that this does not fall into any homework or course work for me. It is just my interest.
I need to prove that limit of a constant gives the constant it self. Can some one provide a link? I have exams or I would have searched myself but unfortunately I don't...
Hi, I was looking for a symbol in math that is commonly applied when a limit to a function does not exist. Is there such a symbol? I could not find any.
Hello . I have problems with two exercises .
1.\lim_{t \to 0 } \frac{2v_1-t^2v_2^2}{|t| \sqrt{v_1^2+v_2^2} }
Here, I have to write when this limit will be exist.
2.\lim_{(h,k) \to (0,0) } \frac{2hk}{(|h|^a+|k|^a) \cdot \sqrt{h^2+k^2} }
Here, I have to write for which a \in \mathbb{R}_+ this...