# 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

.

The limit superior of a sequence

x

n

{\displaystyle x_{n}}
is denoted by

lim sup

n

x

n

or

lim
¯

n

x

n

.

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1. ### 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. ### 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. ### 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. ### Stuck at proving a bounded above Subsequence

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}} ##...
5. ### 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. ### Some hypothetical limits for the Star Wars universe?

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...
7. ### 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 x - \sqrt x} {\sqrt x - \sqrt x}}## Those limits can be evaluated by letting ##x = t^2##...
8. ### 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. ### I Is there a limit to how hot something can get, and if so why?

Question in the title .
10. ### Evaluating This limit

<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 ...
11. ### 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. ### 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. ### 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...

25. ### 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. ### 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. ### 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. ### 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. ### 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...