## Calculus (3rd Edition)

$$\lim _{x \rightarrow 0} x\left[\frac{1}{x}\right]=1$$
Let $y$ be any real number. From the definition of the greatest integer function, it follows that $y-1\lt [y] \leq y$, with equality holding if and only if $y$ is an integer. If $x \neq 0,$ then $\frac{1}{x}$ is a real number, so $$\frac{1}{x}-1\lt \left[\frac{1}{x}\right] \leq \frac{1}{x}$$ Upon multiplying this inequality through by $x,$ we find $$1-x\lt x\left[\frac{1}{x}\right] \leq 1$$ Since $$\lim _{x \rightarrow 0}(1-x)=\lim _{x \rightarrow 0} 1=1$$ Thus, it follows from the Squeeze Theorem that $$\lim _{x \rightarrow 0} x\left[\frac{1}{x}\right]=1$$