Given a loan amount *L* and interest rate *r*
$
x(0)=L
$
Compound yearly:
$
x(1)=(1+r)L=(1+r)\,x(0)
$
Compound monthly:
$
x(1)=\left(1+\frac{r}{12}\right)^{12}x(0)
$
$
\left(1+\frac{r}{12}\right>^{12}1+r
$
As compound period approaches zero:
$
\begin{align}
x(1)&=\lim_{n\rightarrow\infty}\left(1+\frac{r}{n}\right)^nx(0)\\
&= \boxed{e^rx(0)}
\end{align}
$