Arnold on proof of uniqueness

Question

In his proof of Uniqueness, Arnold mentions the integral approaches infinity as x3 approaches x2. How does he come about this conclusion?

Answer 1

Assuming we know that $x_1 \lt x_3 \lt x_2$, then

$$\int_{x_1}^{x_3} \frac{d \xi}{k(\xi - x_2)}= \frac 1k \ln \frac{x_3-x_2}{x_1-x_2}.$$

If $x_3 \to x_2$, then the argument of the logarithm approaches $0$.