Evaluating limit of $\frac{x}{\lfloor x\rfloor}$

Question

How do I evaluate $\lim \limits_{x \to \infty}$$\frac{x}{\lfloor x\rfloor}$ ?

Does the limit exist?

I know that for integer values,the limit evaluates to 1

What about non integer values?

Answer 1

$x-1\leq \lfloor x\rfloor \leq x$, so $1\leq\dfrac{x}{\lfloor x\rfloor}\leq\dfrac{x}{x-1}$ for large $x>0$, now taking limit and use Squeeze Theorem.