Use the Sandwich Theorem to evaluate the limit $\lim_{x\to \infty} \frac{\lfloor x\rfloor}{x}$

Question

Where $⌊x⌋$ is the largest integer not exceeding $x$, use the sandwich theorem to evaluate $\lim_{x\to \infty} \frac{⌊x⌋}{x}$ and $\lim_{x\to -\infty} \frac{⌊x⌋}{x}$

I am unsure what functions to sandwich this between in order to prove the limit?

Answer 1

Hint: For $x\in\mathbb{R}$, $$x-1 < \lfloor x\rfloor \leq x.$$

Answer 2

Note that $x-1<\lfloor x\rfloor\leq x$.

Hence, $\frac{x-1}{x}<\frac{\lfloor x\rfloor}{x}\leq \frac{x}{x}=1$.

It follows that we can take limits to form a sandwich, $\lim_{x\rightarrow a}\frac{x-1}{x}<\lim_{x\rightarrow a}\frac{\lfloor x\rfloor}{x}\leq \lim_{x\rightarrow a}1$.

Taking $a\rightarrow\pm\infty$ would result in a limit of 1.