The universal property of the univariate polynomial ring: For any commutative ring $A$, any commutative $A$-algebra $B$ and any $x\in B$, there exists one and only one $A$-algebra homomorphism from the polynomial ring $A\left[X\right]$ to $B$ which maps $X$ to $x$.
This is the so-called evaluation homomorphism at $x$. I denote this homomorphism by $\lim\limits_{X\to x}$. This has the advantage that we have $\lim\limits_{X\to 0}\dfrac{\left(X+1\right)^n-1}{X}=n$ and similar properties hold just as in classical analysis. The polynomial $\dfrac{\left(X+1\right)^n-1}{X}$ is well-defined (since $X$ is not a zero divisor in $A\left[X\right]$ and divides $\left(X+1\right)^n-1$), but if we would blindly replace $X$ by $0$ we would obtain a $\dfrac{0}{0}$ error.
