justification of a limit I encountered something interesting when trying to differentiate $F(x) = c$.
Consider: $\lim_{x→0}\frac0x$. 
I understand that for any $x$, no matter how incredibly small, we will have $0$ as the quotient. But don't things change when one takes matters to infinitesimals?
I.e. why is the function $\frac0x = f(x)$, not undefined at $x=0$?
I would appreciate a strong logical argument for why the limit stays at $0$. 

A function isn't just an expression, but you can think whether a single expression can be applied to an argument. The expression $0^{-1}$ is rather meaningless, so you don't know how to get the behavior of the function $f(x)=0\cdot x^{-1}$ at $x=0$ from the expression.
Limits are just a way to describe the behavior (if it looks consistent enough that the limit exists) around the point. It doesn't state anything about the value of the function at the point. That is if
$a=\lim_{x\to b}f(x)$
then the function $f_1(x)=\left\{\begin{matrix}f(x)&\text{ if }x\neq b\\a&\text{ if }x=b\end{matrix}\right.$ is continuous.
