3. Given that $f(x)$ is a function defined on the set of real numbers, and $f(x+5)=-f(x)$, when $x \in(5,10)$, $f(x)=\frac{1}{x}$. Then the value of $f(2003)$ is ( ).
(A) $-\frac{1}{8}$
(B) $\frac{1}{5}$
(C) $\frac{1}{3}$
(D) $-\frac{1}{3}$
3.A.

Since $f(x+10)=-f(x+5)=f(x)$, we have
$$
f(2003)=f(200 \times 10+3)=f(3)=-f(8)=-\frac{1}{8} .
$$