All Questions
6
questions
0
votes
1
answer
54
views
Find $a$ such that the limit is zero
Problem :
Let $x>0$ then define :
$$f(x)=\left(\left(\frac{1}{x}\right)!\left(x!\right)\right)^{\frac{1}{x+\frac{1}{x}}}$$
Then find $a$ such that :
$$\lim_{x\to\infty}f(x)-\frac{1}{2}\left(\frac{1}...
5
votes
1
answer
254
views
Evaluate $\lim\limits_{n\to\infty} \frac{\sin(1)+\sin^2(\frac{1}{2})+\ldots+\sin^n(\frac{1}{n})}{\frac{1}{1!}+\frac{1}{2!}+\ldots+\frac{1}{n!}}$
This was a recent problem on the Awesome Math Problem Column. The solution is given as follows:
We shall use Stolz-Cesaro Lemma. We have:
$$\lim_{n\to\infty} \frac{\sin(1)+\sin^2(\frac{1}{2})+\ldots+\...
- 519
3
votes
1
answer
144
views
$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$, for complex variable $z$.
I want to find this limit for complex variable $z$
$$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$$
In the real case I know $\sin(z)$ is bounded by $-1, 1,$ and the limit is $0$. But in the complex case ...
- 261
3
votes
1
answer
204
views
A limit using the Euler number: $\lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$
What is answer of this limit and how can I get it? $c$ and $i$ are constants.
$$\lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$$
I guess it will envolve some Neper/the Euler ...
- 579
1
vote
1
answer
59
views
Finding a limit of a two variable function: $f(x,y)=\frac {\sin(x^2-xy)}{\vert x\vert} $
I have this exercise but not sure if I'm doing it right
$$\lim_{(x,y)\to (0,0)} \frac {\sin(x^2-xy)}{\vert x\vert} $$
I assume $\frac {\sin(x^2-xy)}{\vert x\vert}\le\frac {1}{\vert x \vert} $
then ...
- 11
8
votes
4
answers
540
views
A limit related to super-root (tetration inverse).
Recall that tetration ${^n}x$ for $n\in\mathbb N$ is defined recursively: ${^1}x=x,\,{^{n+1}}x=x^{({^n}x)}$.
Its inverse function with respect to $x$ is called super-root and denoted $\sqrt[n]y_s$ (...
- 47.3k