## Task 12/72

Investigate whether there exists a real number $x$ such that

$$
x=\sqrt{\lim _{n \rightarrow \infty}\left(\frac{1}{n^{3}}+\frac{2}{n^{2}}-\frac{1}{n}\right) \cdot \cot \frac{1}{n}}
$$
A real number $x$ with the required property exists if and only if

$$
0 \leq \lim _{n \rightarrow \infty}\left(\frac{1}{n^{3}}+\frac{2}{n^{2}}-\frac{1}{n}\right) \cot \frac{1}{n}<\infty
$$

holds. We thus investigate the limit. To do this, we set $\frac{1}{n}=z$, so $n=\frac{1}{z}$. As $n \rightarrow \infty$, $z \rightarrow 0$. The limit then takes the form

$$
\lim _{z \rightarrow 0}\left(z^{3}+2 z^{2}-z\right) \cot z
$$

We rearrange:

$$
\begin{aligned}
& \lim _{z \rightarrow 0}\left(z^{3}+2 z^{2}-z\right) \cot z=\lim _{z \rightarrow 0}\left(z^{2}+2 z^{2}-1\right) \frac{z}{\sin z} \cos z= \\
= & \lim _{z \rightarrow 0}\left(z^{2}+2 z^{2}-1\right) \cdot \lim _{z \rightarrow 0} \frac{z}{\sin z} \cdot \lim _{z \rightarrow 0} \cos z=(-1) \cdot 1 \cdot 1=-1
\end{aligned}
$$

The (both necessary and sufficient) condition that the limit is a non-negative real number is thus not satisfied. This means, however, that there is no real number $x$ with the required property.