# Tagged Questions

Questions on the evaluation of limits.

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### Final Question(For Now) On Basics Limit Arithmetics

Is this claim true or false? Given $\lim \limits_{n\to \infty}\ (a_{2n}-a_n)=0$ then $\lim \limits_{n\to \infty}\ a_n$ exists. Thanks a lot.
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### Yet Another Question On Using Basics Limit Arithmetics

Is this claim true? Given $\lim \limits_{n\to \infty}\ a_n=\frac{1}{2}$ Then $\lim \limits_{n\to \infty}\ (a_n - [a_n])=\frac{1}{2}$ I think it's true, but probably I just didn't find the right ...
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### Limits notation

I'm wondering what is the difference in the use of $$\lim\limits_{x \downarrow a}$$ $$\lim\limits_{x \searrow a}$$ $$\lim\limits_{x \nearrow a}$$ $$\lim\limits_{x \uparrow a}$$ I see them ...
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### Help in evaluating $\lim_{x \rightarrow \infty} \frac{1000^x}{x^x} = 0$

I suspect that $$\lim_{x \to \infty} \frac{1000^x}{x^x} = 0.$$ However, I do not know how to prove that this is the case. Any help would be greatly appreciated.
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### Limit at infinity of a uniformly continuous integrable function [duplicate]

Possible Duplicate: $f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$ This is an exercise from Berkeley preliminary exams, Fall 1983 Let ...
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### Big-O, asymptotical dominance, asymptotical equivalence

Let $f(x)= 5x^3+x.$ A) I'm just learning the Big O notation, and my study materials indicate that since $f(x)$ is $O(x^3),$ $f(x)$ is asymptotically dominated by $x^3.$ B) On the other hand, I know ...
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### Why doesn't integrating infinitesimally small likelihoods work in this sense?

We know that something is going to happen after $x$ amount of time, but the exact time at which the event occurs is random within $x$ time. (Like, say we did it a bunch of times where it happened in ...
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### How to prove that $\lim_{\lambda \rightarrow 0} f*h_\lambda (x)=f(x)$ a.e.

Assume that $h_\lambda(x)=\frac{1}{\pi} \frac{\lambda}{\lambda^2+x^2}$, for $\lambda>0$, $x \in \mathbb{R}$. I know that if $f\in L^p$ then $\lim_{\lambda \rightarrow 0} \|f*h_\lambda -f\|_p =0$, ...
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### Convergence of $\sum_{n=1}^{\infty} \frac{n+5}{\sqrt[3]{n^7+n^2-n}}$

Convergence of $$\sum_{n=1}^{\infty} \frac{n+5}{\sqrt[3]{n^7+n^2-n}}$$ I think ratio test will be very tedious, root test too since its not a exponential equation. So I tried limit comparison test, ...
### If $x_1=5$, $x_{n+1}=x_n^2-2$, find $\lim x_{n+1}/(x_1\cdots x_n)$
If $$\left\{x_{n}\right\}\mid x_{1}=5,x_{n+1}=x_{n}^{2}-2,\forall n\geq 1$$ find $$\lim_{n\to\infty}\frac{x_{n+1}}{x_{1}x_{2}\cdots x_{n}}.$$ If someone could help me out with tags, it'd be lovely. I ...