Example 13. Let $a, b, c$ be distinct integers from 1 to 9. What is the largest possible value of $\frac{a+b+c}{a b c}$? (1992, 1st Dannevirke-Shanghai Friendship Correspondence Competition)
Let $P=\frac{a+b+c}{a b c}$.
In equation (1), let $a, b$ remain unchanged temporarily, and only let $c$ vary, where $c$ can take any integer from 1 to 9. Then, from $P=\frac{a+b+c}{a b c}=\frac{1}{a b}+\frac{a+b}{a b c}$, we know that when $c=1$, $P$ reaches its maximum value. Therefore, $c=1$.
Thus, $P=\frac{a+b+1}{a b}$.
In equation (2), let $a$ remain unchanged temporarily, and only let $b$ vary, where $b$ can take any integer from 2 to 9. Then, from $P=\frac{a+b+1}{a b}=\frac{1}{a}+\frac{a+1}{a b}$, we know that when $b=2$, $P$ reaches its maximum value. Therefore, $b=2$.
Thus, $P=\frac{1}{2}+\frac{3}{2 a}$.
In equation (3), $P$ reaches its maximum value when $a$ takes its minimum value, and $a$ can take any integer from 3 to 9, i.e., $a=3$.

Therefore, when $a=3, b=2, c=1$ (the letters can be interchanged), $P$ reaches its maximum value of 1.

Regarding the application of function extremum problems, there is another important aspect, which is the application of function extremum. As long as we master the basic methods of finding the extremum of a function, such problems are not difficult to solve.