Articles of limits

Limit with square and cube root difference $\lim_{n \to \infty} \left(\sqrt{n^2 + n} – \sqrt{n^3 + n^2}\right)$

I’ve been banging my head at this one for the last hour and I can’t seem to find the solution. I’ve read through almost all of the limit problems that were asked to be solved here and the tricks used just don’t work on this one, or maybe I’m just not seeing it. $\lim_{n \to […]

How prove this sequences have limts and find this value?

let $f:R\longrightarrow R$,and $f(x)=x-\dfrac{x^2}{2}$,and $$L_{1}(x)=f(x),L_{2}(x)=f(f(x)),L_{3}(x)=f(L_{2}(x)),\cdots,L_{n}(x)=f(L_{n-1}(x))$$ and let $$a_{n}=L_{n}\left(\dfrac{17}{n}\right)$$ prove that $$\displaystyle\lim_{n\to\infty}\left(na_{n}\right)$$ is exsit,and find the value limit. my idea: I want find this $$L_{n}(x)=\cdots?$$ But is very ugly,becasuse $$L_{2}(x)=f(f(x))=\left(x-\dfrac{x^2}{2}\right)-\dfrac{(x-\dfrac{x^2}{2})^2}{2}=-\dfrac{1}{8}x^4-\dfrac{1}{2}x^3-x^2+x$$

converging subsequence on a circle

We know that any sequence on $S^1$ must have a converging extracted subsequence, as $S^1$ is compact. Now, consider the sequence $a_n=(\cos(n),\sin(n))$. Could you find explicitly a subset of the natural numbers such that the corresponding subsequence converges? I don’t even know whether it is possible to work it out, or whether there exists a […]

Need a hint to evaluate $\lim_{x \to 0} {\sin(x)+\sin(3x)+\sin(5x) \over \tan(2x)+\tan(4x)+\tan(6x)}$

I know that $\sin A + \sin B + \sin C = 4\cos({A \over 2})\cos({B \over 2})\cos({C \over 2})$ when $A+B+C=\pi$. If ${x \to 0}$ then I have a half circle, right? If it is right then I have $\tan(2x) + \tan(4x) + \tan(6x)=\tan(2x)\tan(4x)\tan(6x)$. I got stuck at $${4\cos({x \over 2})\cos({3x \over 2})\cos({5x \over 2}) […]

Fatou's Lemma and Almost Sure Convergence (Pt. 1)

I have a question regarding Fatou’s Lemma and a sequence of random variables converging almost surely. Fatou’s Lemma states If $\forall n \in \mathbb{N}, \,\, X_{n} \ge 0$ and $\displaystyle X = \liminf_{n \rightarrow \infty} X_{n}$, then $\displaystyle\mathbb{E}[ \liminf_{n \rightarrow \infty}\: X_{n}] \le \liminf_{n \rightarrow \infty}\: \mathbb{E}[ X_{n}]$ Suppose we also know that $X_{n} \rightarrow […]

Finding $\lim_{x \to 0} \frac {a\sin bx -b\sin ax}{x^2 \sin ax}$ witouth L'Hopital, what is my mistake?

I was working on this question. $\lim_{x \to 0} \dfrac {a\sin bx -b\sin ax}{x^2 \sin ax}$ $\lim_{x \to 0} \dfrac {1}{x^2} \cdot \lim_{x \to 0} \dfrac { \frac {1}{abx}}{\frac {1}{abx}} \cdot \dfrac {a\sin bx -b\sin ax}{\sin ax}$ $\lim_{x \to 0} \dfrac {1}{x^2} \cdot \dfrac {\lim_{x \to 0} \frac {\sin bx}{bx}- \lim_{x \to 0}\frac{\sin ax}{ax}}{\frac 1b […]

How to compute this limit $\lim_{n\to ∞}\frac{1}{n}\log{{n\choose 2\alpha n}}$

$$\lim_{n\to ∞}\frac{1}{n}\log{{n\choose 2\alpha n}}=\frac{3}{2}((1-2\alpha) \log{2\alpha}+2\alpha\log2\alpha)$$ such that $2\alpha n\le n$ I tried to use Stirling formula and we get $$\lim_{n\to ∞}\frac{1}{n}\log{{n\choose 2\alpha n}}=\lim_{n\to ∞}\frac{1}{n}\log\frac{n^{\frac{3n}{2}}}{2\pi(n-2\alpha n)^{\frac{3((n-2\alpha n)}{2}{(2\alpha n)}^{3\alpha n}}}=$$ $$=\lim_{n\to ∞}\log{\frac{n^{\frac{3}{2}}}{2\pi(n-2\alpha n)^{\frac{3((1-2\alpha )}{2}{(2\alpha n)}^{3\alpha }}}}$$ but I couldn’t continue

Evaluate $\lim_{n\to\infty}\prod_{k=1}^{n}\frac{2k}{2k-1}$

How can I calculate the following limit: $$ \lim_{n\to \infty} \frac{2\cdot 4 \cdots (2n)}{1\cdot 3 \cdot 5 \cdots (2n-1)} $$ without using the root test or the ratio test for convergence? I have tried finding an upper and lower bounds on this expression, but it gives me nothing since I can’t find bounds that will […]

The $n$th integral of $\ln(x)$ and fractional derivatives

For a related question, I need to know the $n$th integral of $\ln(x)$ and the fractional derivative of $\ln(x)$. A break down of how fractional derivatives may be found on the Wikipedia. In particular, I need to calculate $\frac{d^{1/2}}{dx^{1/2}}\ln(x)$ and $\frac{d^{-n}}{dx^{-n}}\ln(x)$ where that is the $n$th integral of $\ln(x)$. The fractional derivative in this scenario […]

Show that $\lim_{x \to c} x^{3}=c^{3}$

Please check my proof I will use the fact that $\lim_{x \to c}x^{3}=c^{3}$ equivalent $\lim_{x \to c}x\lim_{x \to c}x\lim_{x \to c}=(c)(c)(c)$ for $\lim_{x \to c} x=c$ we let $\epsilon >0$ and $\delta >0$ we have $0<|x-c|<\delta \leftrightarrow |x-c|<\sqrt[3]{\epsilon }$ but in this case we want to proove $\lim_{x \to c}x^{3}=c^{3}$ We will have $0<|x-c|<\delta \leftrightarrow […]