# 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. ### 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

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

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

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5. ### I Limit of (a^n)/n for a>1

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6. ### Some hypothetical limits for the Star Wars universe?

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7. ### B Rationale Behind t-Substitution for Evaluating Limits?

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8. ### Calculus What are some books for learning the techniques of Calculus?

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

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

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11. ### Limit with the quotient law

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12. ### Using a Partial Limit

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13. ### I Precise intuition about limits and infinitesimals

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25. ### I A problematic limit to prove

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

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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

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29. S

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

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30. ### Partial derivative and limits

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