Intereting Posts

Showing $G/Z(R(G))$ isomorphic to $Aut(R(G))$
Is a totally ordered set well-ordered, provided that its countable subsets are?
Is there a general formula for the derivative of $\exp(A(x))$ when $A(x)$ is a matrix?
Examples of group extension $G/N=Q$ with continuous $G$ and $Q$, but finite $N$
If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
Can we make an integral domain with any number of members?
“belongs to” versus “contained in”
Proof that the real numbers are countable: Help with why this is wrong
$K$-theory of smooth manifolds: continuous vs. smooth vector bundles
Why isn't $e^{2\pi xi}=1$ true for all $x$?
Non-Metrizable Topological Spaces
Congruence between Bernoulli numbers
What is the reasoning behind why the ratio test works?
Irrationality of sum of two logarithms: $\log_2 5 +\log_3 5$
Proof of power rule for limits?

Using only the delta definition of a limit, how can we prove that the sequence $\{a_n\}$, where $a_n = \sin n$, as $n$ tends to infinity does not have a limit?

Thanks!

- Finding continuity and differentiability of a multivariate function
- slope of a line tangent through a point
- How to determine equation for $\sum_{k=1}^n k^3$
- Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?
- an indefinite integral $\int \frac{dx}{\sin{x}\sqrt{\sin(2x+\alpha)}}$
- Differentiate vector norm by matrix

- Find the limit without use of L'Hôpital or Taylor series: $\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2 x}\right)$
- Find the limit of $(2\sin x-\sin 2x)/(x-\sin x)$ as $x\to 0$ without L'Hôpital's rule
- Why does the sum of these trigonometric expressions give such a simple result?
- Looking for a proof of Cleo's result for ${\large\int}_0^\infty\operatorname{Ei}^4(-x)\,dx$
- Optimization with cylinder
- A proof for Landau inequality and similar cases
- How do you calculate this limit without using L'Hopital or Taylor?
- Optimisation Problem on Cone
- Find $\lim_\limits{x\to 0}{x\left}$. Am I correct?
- Evaluate $ \int_{0}^{1} \ln(x)\ln(1-x)\,dx $

No need for $\epsilon$ actually.

If $\sin(n) \rightarrow l$, then $\sin(n+1)$ also, and $\sin(n+1)=\sin(n)\cos(1)+\sin(1)\cos(n)$.

Since both $\sin(n)$ and $\sin(n+1)$ have limit $l$ and $\sin(1) \neq 0$, $\cos(n) \rightarrow \frac{l(1-\cos(1))}{\sin(1)}$, and so $e^{in}=\cos(n)+i \sin(n)$ has a limit.

But $e^{i(n+1)}$ must then have the same limit (call it $x$), which implies $x=e^{i} x$, and since $e^{i} \neq 1$, $x$ has to be zero, which is a contradiction with the fact that $|e^{in}|=1$.

Assume $\lim \sin(n) = l$. Then so is $\lim \sin(2n) = l$. So $\lim \cos(2n) = 1 – 2l^2$, but so does the limit of $\cos(2(n + 1))$. Now apply the sum-formula to $\sin(2(n + 1) – 2n)$.

The following are true, based on standard trigonometric identities and $\sin(1) \approx 0.84147$ and $\sin(3) \approx 0.14112$:

$$\begin{align}

\textrm{if } \sin(n) \le -0.4, & \textrm{ then } 0 < \sin(n+3) ; \\

\textrm{if } -0.4 \le \sin(n) \le 0.4, & \textrm{ then } \sin(n+1) < -0.4 \textrm{ or } 0.4 < \sin(n+1) ; \\

\textrm{if } 0.4 \le \sin(n),& \textrm{ then } \sin(n+3) < 0;

\end{align}$$

so there is no value $L$ where for any positive $\varepsilon < 0.2$ you have all of $\sin(n), \sin(n+1), \sin(n+3)$ and $\sin(n+4)$ within $\varepsilon$ of $L$.

Let if possible $\sin n\rightarrow x$. Then $\sin k=\sin(n+k-n)=\sin(n+k)\cos k-\cos(n+k)\sin n\rightarrow x(\cos k-\sqrt{1-x^2})$ for each positive integer k. Now as $k\rightarrow \infty$ implies that $x=x.0=0$ which shows that $\sin k=0$ for all k…a contradiction.

- Finding coefficient of a complicated binomial expression?
- What is the square of summation?
- Is it true that $aH = bH$ iff $ab^{-1} \in H$
- If $x$ is real and $p=\frac{3(x^2+1)}{2x-1}$, prove that $p^2-3(p+3)\ge0$
- Infinite group with only two conjugacy classes
- Finding all complex zeros of a high-degree polynomial
- Continuity of a function of product spaces
- If $\frac{z^2_{1}}{z_{2}z_{3}}+\frac{z^2_{2}}{z_{3}z_{1}}+\frac{z^2_{3}}{z_{1}z_{2}} = -1.$Then $|z_{1}+z_{2}+z_{3}|$
- Shift Operator has no “square root”?
- $C(S^1)$ does not have a single generator
- A monotonically increasing series for $\pi^6-961$ to prove $\pi^3>31$
- Proof that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then $p$ can't be written as a sum of two squares
- Confusion about the null (empty) set being contained in other sets
- Proving that $X/R$ is Hausdorff $\implies$ $R$ closed.
- Sampling $Q$ uniformly where $Q^TQ=I$