# Tagged Questions

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### Limit for $\lim _{n \to \infty}(n+2)^{2}\sin\frac{1}{n}$

Can't prove the limit $$\lim_{n \to \infty}(n+2)^{2}\sin\frac{1}{n}=\infty.$$ by definition it should start: Let $M>0$. There exists an $N>0$ for every $n>N$: ...
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### Convergence of sequence to arbitrary large number

Suppose arbitrary positive $a<\infty$. Let $a_n=O(n^b)$ for some $b>0$. What is the convergence rate of $a_n$ to $a$? Can I say $\max(a-a_n,0)=O(a/n^b)$? If not, what can I deduce and why?
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### Question about $f$ continuous function with these conditions?

Suppose I have a differentiable and bounded function $$f: [0, + \infty) \longrightarrow \mathbb{R}$$ such that $$\forall x \in [0, + \infty) \, : f(x) \cdot f'(x) > \sin x.$$ The question is: ...
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### $\frac{1}{\infty}$ - is this equal $0$?

I've seen that wolfram alpha says: $$\frac{1}{\infty} = 0$$ Well, I'm sure that: $$\lim_{x\to \infty}\frac{1}{x} = 0$$ But does $\frac{1}{\infty}$ only makes sense when we calculate it's limit? ...
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### Limit Involving Factorials

How would you go about calculating $$\lim_{x \to \infty} \frac{x!}{(x - k)!}$$ for some constant $k > 0$?
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### How does $1^\infty=\infty$?

I remember hearing in school long ago that $1^\infty=\infty$. I was just wondering if anyone could explain this in laymen's terms?
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### negative and positive infinity

This is a weird question that I thought of and I was wondering if I could get some help. So normally $\frac{1}{x} = \frac{1}{y}$ then x and y would have to both be the same number, but with infinity ...
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### Counting backwards from $\infty$ and negative cardinality

I was thinking of this different definition of natural numbers and weird way of counting, similar to Peano, but counting backwards from infinity, not forward from zero. I tried to answer the question: ...
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### Finding a limit using arithmetic over cardinals

What is the value of: $$\lim_{n \to \infty} \frac{n}{2^n} (n \in \mathbb{N})$$ It seems to me that I can use L'Hopital's rule, but does that rule take into account types of infinity? More precisely, ...
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### Infinite square-rooting

$\lim_{n\to\infty} {\sqrt{1+{\sqrt{2+{\sqrt{\cdots +\sqrt{n}\ }\ }\ }\ }\ \ }\ } = ?$ Either closed answer or an upper bound would help.
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### Evaluation of a limit

Here is a question on limits. I would like to ask help. Here it goes: $$\lim_{N\to\infty}\left(\frac{\sum_{j=0}^{N}\left(\frac{j}{N}\right)^{n+1}}{\sum_{j=0}^{N}\left(\frac{j}{N}\right)^{n}}\right)$$ ...
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### Which sign does $\lim_{n\to\infty}(-2)^n$ have?

How can I express $\lim_{n\to\infty}(-2)^n$ using $\infty$? Which sign does it have, plus or minus?