11.2. Calculate the limit $\lim _{n \rightarrow \infty}\left(\left(1+\frac{1}{2}\right) \cdot\left(1+\frac{1}{2+3}\right) \cdot \ldots \cdot\left(1+\frac{1}{2+3+\ldots+n}\right)\right)$.
Solution.

Let $P_{n}=\left(1+\frac{1}{2}\right) \cdot\left(1+\frac{1}{2+3}\right) \cdots \cdot\left(1+\frac{1}{2+3+\ldots+n}\right)$.

$$
\begin{aligned}
& S_{n}=1+2+\ldots+n=\frac{n(n+1)}{2}, \forall n \in \mathbb{N}^{*} \Rightarrow \\
& S_{n}-1=2+3+\ldots+n=\frac{n(n+1)}{2}-1=\frac{n^{2}+n-2}{2}=\frac{(n-1)(n+2)}{2}, \forall n \in \mathbb{N}^{*} \Rightarrow \\
& P_{n}=\frac{S_{2}}{S_{2}-1} \cdot \frac{S_{3}}{S_{3}-1} \cdot \ldots \cdot \frac{S_{n}}{S_{n}-1}=\prod_{k=2}^{n} \frac{S_{k}}{S_{k}-1}=\prod_{k=2}^{n} \frac{k(k+1)}{(k-1)(k+2)}=
\end{aligned}
$$

$$
\begin{gathered}
=\frac{\left(\prod_{k=2}^{n} k\right) \cdot\left(\prod_{k=3}^{n+1} k\right)}{\left(\prod_{k=1}^{n-1} k\right) \cdot\left(\prod_{k=4}^{n+2} k\right)}=\frac{n!\cdot \frac{(n+1)!}{1 \cdot 2}}{(n-1)!\cdot \frac{(n+2)!}{1 \cdot 2 \cdot 3}}=\frac{3 \cdot n!\cdot(n+1)!}{(n-1)!\cdot(n+2)!}=\frac{3 \cdot n}{n+2}, \forall n \in \mathbb{N}^{*} \Rightarrow \\
\lim _{n \rightarrow \infty} P_{n}=\lim _{n \rightarrow \infty} \frac{3 \cdot n}{n+2}=3 .
\end{gathered}
$$

11. 3. Determine all continuous functions $f: \mathbb{R}_{+}^{*} \rightarrow \mathbb{R}$, which satisfy the relation

$$
f\left(\frac{x}{y}\right)=\frac{f(x)}{y}-\frac{f(y)}{x}, \quad \forall x, y \in \mathbb{R}_{+}^{*}
$$