2. Given two non-zero numbers. If you add one to each of them, as well as subtract one from each of them, then the sum of the reciprocals of the four resulting numbers will be equal to 0. What number can result if you subtract the sum of their reciprocals from the sum of the original numbers? Find all possibilities. (S. Berlov)
Answer: 0. Solution: Let the given numbers be a and b. We transform the sum of the four numbers given in the condition: $0=\frac{1}{a+1}+\frac{1}{a-1}+\frac{1}{b+1}+\frac{1}{b-1}=\frac{2 a}{a^{2}-1}+\frac{2 b}{b^{2}-1}$. Let $x=\frac{a}{a^{2}-1}=-\frac{b}{b^{2}-1}$. Then

$$
a-\frac{1}{a}+b-\frac{1}{b}=\frac{a^{2}-1}{a}+\frac{b^{2}-1}{b}=\frac{1}{x}-\frac{1}{x}=0
$$