Does the following limit exist? If so, determine it.

$$
\lim _{x \rightarrow 0} x \cdot\left[\frac{1}{x}\right]
$$
We will show that the limit in question exists and its value is 1. Indeed, for any number $z$, we obtain the inequality

$$
z-1x\left[\frac{1}{x}\right] \geqq 1
$$

In both cases,

$$
\left|x\left[\frac{1}{x}\right]-1\right|<|x|
$$

thus the difference between the function value and the supposed limit is less than $\varepsilon$ if $|x|<\varepsilon$.

Jenő Pál (Kaposvár, M. Táncsics Gymnasium, 3rd grade)