limit Definition and Topics - 86 Discussions

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




x

n




{\displaystyle x_{n}}
is denoted by





lim sup

n





x

n




or





lim
¯



n






x

n


.


{\displaystyle \limsup _{n\to \infty }x_{n}\quad {\text{or}}\quad \varlimsup _{n\to \infty }x_{n}.}

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  1. L

    I Prove that the limit of this matrix expression is 0

    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...
  2. R

    I Infinite limit definition

    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 $$\lim_{x\to p^-}=-\infty? $$
  3. C

    I ##(a_n) ## has +10,-10 as partial limits. Then 0 is also a partial limit

    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...
  4. C

    Stuck at proving a bounded above Subsequence

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  5. Delta2

    I Limit of (a^n)/n for a>1

    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...
  6. M

    Some hypothetical limits for the Star Wars universe?

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  7. bagasme

    B Rationale Behind t-Substitution for Evaluating Limits?

    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##...
  8. Adesh

    Calculus What are some books for learning the techniques of Calculus?

    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...
  9. A

    I Is there a limit to how hot something can get, and if so why?

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  10. navneet9431

    Evaluating This limit

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  11. A

    Limit with the quotient law

    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.
  12. navneet9431

    Using a Partial Limit

    <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...
  13. S

    I Precise intuition about limits and infinitesimals

    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...
  14. D

    Approximations with the Finite Square Well

    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 \\...
  15. V

    Limit of the form ∞-∞

    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...
  16. V

    Limit of the form 0/0

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  17. E

    Computations with limits

    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...
  18. D

    I Limit of an extension

    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...
  19. EEristavi

    Continuity of Function - f(x)=|cos(x)|

    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...
  20. akaliuseheal

    L'Hopital's rule

    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...
  21. H

    Find the limit as h --> 0 for this trigonometery equation

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  22. R

    Limit of x^α.sin²(x!)/(x+1) as x approaches infinity is?

    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...
  23. R

    Does x sin(1/x) exist as x approaches zero?

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  24. A

    Finding the limit at infinity

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  25. D

    I A problematic limit to prove

    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...
  26. H

    Find limit x to infinity from f(x) contains squareroot of x

    Homework Statement ##\lim x \to \infty \frac{\sqrt{x+1} - \sqrt{x}}{\sqrt{3x + 5} - \sqrt{3x + 1}}## Homework Equations The Attempt at a Solution ##\lim x \to \infty {\sqrt{x+1} - \sqrt{x}} * \lim x \to \infty \frac{1}{\sqrt{3x + 5} - \sqrt{3x + 1}}## ##\lim x \to \infty \frac{(x+1) -...
  27. pawlo392

    A Lebesgue measure and integral

    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.
  28. shihab-kol

    B Proof of a limit rule

    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...
  29. S

    B Is there a particular symbol in Math for inexisting limits?

    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.
  30. pawlo392

    Partial derivative and limits

    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...
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