I solved this limit problem by following this way, but I'm not exactly sure about ....
can anyone help me and tell me if it is correct? the problem is:
Let $k>1$. If it exists, calculate the limit of the sequence $(x_n)$, $x_n := \Biggl(k \sin (\frac{1}{n^2}) + \frac{1}{k}\cos n \Biggr)^n.$
my solution:
From trigonometry we know that: $ 0< \sin x < x <\tan x$ for $0 < x < \pi/2 ;$ then $\sin x < x \Rightarrow \sin \frac{1}{n^2}<\frac{1}{n^2};$ the cosine function we know to be a bounded function; then: $\left | \cos x \right |\le1\Rightarrow -1\le\cos x\le 1 \Rightarrow -\frac{1}{k}\le \frac{1}{k}\cos n \le\frac{1}{k} $ so we have that: $k \sin (\frac{1}{n^2}) + \frac{1}{k}\cos n \le \Biggl(k\frac{1}{n^2}+\frac{1}{k}\Biggr) \Rightarrow\Biggl(k \sin (\frac{1}{n^2}) + \frac{1}{k}\cos n \Biggr)^n\le \Biggl(k\frac{1}{n^2}+\frac{1}{k}\Biggr)^n,$ then $x_n \le \Biggl(\frac{k^2+n^2}{kn^2}\Biggr)^n;$ On the other hand, we have also that: $\left | \sin x \right |\le1\Rightarrow -1\le\sin x\le 1 \Rightarrow -{k}\le\ k \cdot \sin (\frac{1}{n^2}) \le{k},$ and then in particular $-\frac{1}{k}\le\cos n \qquad,\qquad -{k}\le\ k \cdot \sin (\frac{1}{n^2});$ well: $\Biggl(-k-\frac{1}{k}\Biggr)\le\Biggl(k \sin (\frac{1}{n^2}) + \frac{1}{k}\cos n \Biggr) \Rightarrow -\Biggl(k+\frac{1}{k}\Biggr)^n\le\Biggl(k \sin (\frac{1}{n^2}) + \frac{1}{k}\cos n \Biggr)^n,$ end then: $-\Biggl(k+\frac{1}{k}\Biggr)^n\le x_n;$ combining the two results we obtained: $-\Biggl(k+\frac{1}{k}\Biggr)^n\le x_n\le\Biggl(\frac{k^2+n^2}{kn^2}\Biggr)^n.$ at this point we have found that $a_n$ $b_n$ such that: $a_n \le x_n \le b_n,$ an so $x_n$, if it admits limit, must be admitted to the same limit $a_n$ and $b_n$; so: $\lim_{n\to \infty}a_n=\lim_{n\to \infty}-\Biggl(k+\frac{1}{k}\Biggr)^n=-\lim_{n\to \infty}\Biggl(\frac{k^2+1}{k}\Biggr)^n=-\infty$
$\lim_{n\to \infty}b_n=\lim_{n\to \infty}=\Biggl(\frac{k^2+n^2}{kn^2}\Biggr)^n=\Biggl[\frac{n^2\cdot(1+\frac{k}{n^2})}{kn^2}\Biggr]^n =\frac{\Biggl[(1+\frac{k}{n^2})^{n^2}\Biggr]}{k^n}^\frac {1}{n}=\frac {e^\frac {k}{n}}{k^n}=\frac {1}{k^n}=0$
as seen , the two limits are different, so the sequence $ x_n $ admits no limit. In conclusion:
$\lim_{n\to \infty}\Biggl(k \sin (\frac{1}{n^2}) + \frac{1}{k}\cos n \Biggr)^n=\nexists$