# Questions tagged [limsup-and-liminf]

For questions concerning the definition and properties of limit superior and limit inferior of sequences of sets or real numbers.

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### How $\limsup_{n\to\infty}\left[\frac{a_n}{n^\varepsilon}\right]^{1/\log\log n}=e^{1+\varepsilon}\implies a_n=O(n^\varepsilon\log^{1+\varepsilon}n)$?

Question Let $(a_n) \subseteq \mathbb R_+$ be a sequence of non-negative numbers such that $$\limsup_{n \to \infty} \left[ \frac{a_n}{n^\varepsilon} \right]^{1/\log \log n} = e^{1+\varepsilon}$$ for ...
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### Can we use power mean to generalize min and max for complex numbers?

Power mean $M_p(a,b)$ of order $p \in \mathbb{R}$ for a pair $(a,b) \in \mathbb{R}^+$ is defined as $M_p(a,b)= \Big(\frac{a^p+b^p}{2}\Big)^{\frac{1}{p}}$. For example $p = 1$ gives arithmetic mean, ...
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### How can I solve this without L'Hôpital's rule or Taylor series? [closed]

How can I solve this limit without L'Hôpital's rule or Taylor series?$$\lim_{x\to -1}\frac{\sin(x^3-x)}{x+1}.$$ I was trying to solve this limit but I'm stuck when I multiply it by conjugate of the ...
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### "Properties" of the definitions of limit superior and inferior

I was working on a proof about measure theory, where I was asked to show that for any sequence of subsets $\left(A_{n}:n\in \mathbb{N}\right)$ of some set $X$, if we created another sequence of ...
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### $\lim\sup$ and $\lim\inf$ of a sequence of independent random variables with two states

Let $(X_n)_{n\geq1}$ be a sequence of random variables such that $$\mathbb{P}(X_n= n^{2/3})=1-\mathbb{P}(X_n=0)=\frac{1}{3n}$$ Find $\lim\sup X_n$ and $\lim\inf X_n$. We know that \begin{align} \{\...
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### (When) do students typically learn about limit superior and inferior?

I've been taking the Khan AP Calculus BC course, and I also have been following through with pretty extensive calculus notes found here. In neither of these (or when I learned Calc in high school) ...
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### Does the limit : $\lim _{x\to \infty }\frac{\ln x^{\frac{1}{3}}}{\sin x}$ exist

I tried to evaluate the limit $$\lim _{x\to \infty }\frac{\ln x^{\frac{1}{3}}}{\sin x}$$ but came to the conclusion that the limit would not exist as $\sin x$ changes sign too frequently so obviously ...
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### A nonnegative, integrable, Lipschitz function $f$ satisfies $\lim \inf_{n \rightarrow \infty} \sqrt{n}f(n) = 0$

Let $f$ be a nonnegative integrable and Lipschitz function in $\mathbb{R}$ with Lipschitz constant $C$. Prove that $\lim \inf_{n \rightarrow \infty} \sqrt{n}f(n) = 0.$ The issue I have with this ...
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This is follow up question of the following two questions I asked Sequencial chracterisation of $\limsup f(x)$ Show equivalence of definitons of $\limsup f(x)$ I would like to show that there is $\... • 754 2 votes 0 answers 69 views ### Show equivalence of definitons of$\limsup f(x)$This is a follow up question to the following question I asked. Sequencial chracterisation of$\limsup f(x)$. I would like to pove that is equivalent to Proof attempt "$\Leftarrow$" Let$\...
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Let {$a_n$} and {$b_n$} be two sequences of real numbers such that {$b_n$} converges to L. It is known that the supremum S of the set {$a_n$: n $\in$ N} exists and is not equal to any of the $a_j$'s. ...