Exponential limit
= Exponential limit
{title2=$\lim_{n\to\infty}(1+x/n)^n=e^x$}
For fixed real $x$, the base $1+x/n$ is positive for sufficiently large $n$. Applying <Taylor's theorem> to the <natural logarithm> at one gives $n\log(1+x/n)=x+O(n^{-1})$. Continuity of the <exponential function> then proves the displayed <limit>. In particular $(1-z/n)^n\to e^{-z}$ for fixed $z\geq0$, a basic source of <exponential distribution> limits in <probability theory>.