what is $\lim\limits_{x\to 0} \lceil x\rceil/x$? What is $\displaystyle \lim_{x \to 0} \frac{\lceil x \rceil}{x}$ ?  Here, $\lceil x \rceil$ is the ceiling function at $x$.

For left limit and right limit as $x\to 0$.
Assuming that [x] is the floor of x, then look at this graph.

Assuming that [x] is the ceiling of x, then look at this similar looking graph.

Assuming that [x] is the "nearest integer" function, then consider what the nearest integer is on the interval [-0.49, 0.49].  

If this is something else, please specify.
$\lim \limits_{x \to 0} \frac{\lceil x \rceil}{x}$

This is what the Alpha Wolf says:


  $\lim \limits_{x \to 0^-} \frac{\lceil x \rceil}{x} = 0$
  
  $\lim \limits_{x \to 0^+} \frac{\lceil x \rceil}{x} = \infty$


The two limits aren't equal. And what could this possible mean?

How did we get this? Look here:


  $\lim \limits_{x \to 0^-} \lceil x \rceil = 0$
  
  $\lim \limits_{x \to 0^+} \lceil x \rceil = 1$


And:


  $\lim \limits_{x \to 0^-} x = 0$
  
  $\lim \limits_{x \to 0^+} x = 0$