  Let $f(x)$ be a real function which has $(n+1)$-th derivative on an interval
$[a, b]$. For any point $x_0\in (a, b)$ and any integer $0\leq k\leq n$, denote
by $S_{k,x_0}(x)$ the $k$-th truncation of the Taylor expansion of $f(x)$ at
$x_0$, i.e. $$S_{k,x_0}(x)=\sum_{i=0}^k\frac{f^{(i)}(x_0)}{i!}(x-x_0)^i.$$ In
this note, we consider the $L_2$-approximation of $f(x)$ by polynomials of
degree $\leq k$, we show that $S_{k,x_0}(x)$ is the limit of the best
approximations of $f(x)$ on $[x_0-\varepsilon, x_0+\varepsilon]$ as
$\varepsilon\to 0$.
