query: show that for any irrational $\alpha $ the limit $\lim _{n \to\infty} \sin n \alpha \pi $ does not exist ..
positive_passages.docid: 150917
positive_passages.text: Show that $\lim_{n\to \infty}\sin{n\pi x} =0$ if $x\in \mathbb{Z},$ but the limit fails to exist if $x\notin \mathbb{Z}.$
negative_passages.docid: 13592
negative_passages.text: How can $\pi$ be irrational if it can be calculated using an infinite series of rational numbers?
negative_passages.docid: 152640
negative_passages.text: show that $\lim(\pi(x)/x) = 0$
negative_passages.docid: 129697
negative_passages.text: Prove that a limit doesn't exist
negative_passages.docid: 79419
negative_passages.text: Is $\sin \infty$ an indeterminate form?
negative_passages.docid: 125812
negative_passages.text: Is sin(x) necessarily irrational where x is rational?
negative_passages.docid: 5402
negative_passages.text: Prove or the disprove the existence of a limit of integrals
negative_passages.docid: 97747
negative_passages.text: Limit of $\lim\limits_{n\to\infty} (1 + \frac{x_n}{n})^n$
negative_passages.docid: 16642
negative_passages.text: Showing $a_n=\sin(n)$ does not converge
negative_passages.docid: 13533
negative_passages.text: Proof that the limit of $\frac{1}{x}$ as $x$ approaches $0$ does not exist
negative_passages.docid: 155188
negative_passages.text: How do I prove this limit does not exist: $\lim_{x\rightarrow 0} \frac{e^{1/x} - 1}{e^{1/x} + 1} $
negative_passages.docid: 18882
negative_passages.text: The limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0?
negative_passages.docid: 26109
negative_passages.text: Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$
negative_passages.docid: 148782
negative_passages.text: How to prove these two limits $\lim_{n\to\infty}\frac{a_{n}}{n}$ and $\lim_{n\to\infty}\frac{a_{n+1}-a_{n}}{n}$ exist?
negative_passages.docid: 63418
negative_passages.text: Show that $f_n(x) = \frac{x^2}{n}$ for $x \in \mathbb{R}$ does not converge uniformly towards $0$ for $n \rightarrow \infty$
negative_passages.docid: 94244
negative_passages.text: $\sin \frac{1}{x}$ is not uniformly continuous on $[1, \infty]$
negative_passages.docid: 112481
negative_passages.text: Given $a_{n+1} = \frac{{a_n}}{\sin({a_n})}$ with $0 < a_1 < \frac{\pi}{2}$. Show that the limit $\lim_{n\rightarrow\infty}{a_n}$ exists and find it.