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 $x\to 0$.
 A: 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.
A: No need for graphs or WA.  Just note that
$$
\lceil x \rceil = \left\{ 
\begin{array}{cl}
 1 & x \in (0,1) \\
 0 & x \in (-1,0)
\end{array}
\right.$$
Hence, $\displaystyle\lim_{x \to 0^+} \frac{\lceil x \rceil}{x} = \lim_{x \to 0^+} \frac{1}{x} = + \infty$.  On the other hand, $\displaystyle\lim_{x \to 0^-} \frac{\lceil x \rceil}{x} = \lim_{x \to 0^-} \frac{0}{x} = 0$.
A: What is the value of $\lceil x\rceil$ for small positive $x$? Think of $0<x<1$.
A: $\lim \limits_{x \to 0} \frac{\lceil x \rceil}{x}$
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:

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

A: The limit is equivalent to the derivative of the ceiling function at 0. The derivative of the ceiling function, by observing the graph, is constant when x is not an integer, and undefined when x is an integer. The above limit is therefore undefined. 
A: $$\lim_{x \to 0^+} \frac{\lceil x \rceil}{x} = \frac{\lim_{x \to 0^+}\;\lceil x \rceil}{\lim_{x \to 0^+} \;x} = \frac{1}{\lim_{x \to 0^+} \;x} = \lim_{x \to 0^+}\frac{1}x = \infty$$
$$\lim_{x \to 0^-} \frac{\lceil x \rceil}{x} = \frac{\lim_{x \to 0^-}\;\lceil x \rceil}{\lim_{x \to 0^-} \;x} = \frac{0}{\lim_{x \to 0^-} \;x} = \lim_{x \to 0^-}\frac{0}x = 0$$
$$\lim_{x \to 0^+} \frac{\lceil x \rceil}{x} \not = \lim_{x \to 0^-} \frac{\lceil x \rceil}{x} \therefore \;\, \not \exists \;\;\lim_{x \to 0} \frac{\lceil x \rceil}{x}$$
