TITLE: Division by $0$ and its restrictions
QUESTION [100 upvotes]: Consider the following expression:
$$\frac{1}{2} \div \frac{4}{x}$$
Over here, one would state the restriction as  $x \neq 0 $, as that would result in division by $0$. 
But if we rearrange the expression, then:
$$\begin{align}
\frac12\div\frac4x &= \frac{1}{2} \times \frac{x}{4} \\
&= \frac{x}{8}
\end{align}$$
In this expression, there are no restrictions. If we substitute $x = 0$, then the answer is $\frac{0}{8} = 0$.
So, how come in the first unsimplified expression, when we substitute $ x =0$, we get undefined, whereas in the simplified expression we get $0$?

REPLY [3 votes]: Consider the function
f(x)=x^3/x^4
here if you apply limit and put x=0 then definitely one will get 1/0 form which is undefined.
So make use of the L'Hospitals rule here that is take derivative of numerator and denominator term separately that is,
Applying limit when x approaches to zero and then taking limit