Question:
There are numbers $A$ and $B$ for which  \[\frac A{x-1}+\frac B{x+1}=\frac{x+2}{x^2-1}\] for every number $x\neq\pm1$.  Find $B$.

Answer:
We can approach this problem by picking clever values for $x$.  If $x=-2$ we get  \[\frac A{-2-1}+\frac B{-2+1}=0,\] so \[A+3B=0.\]

If $x=0$ we get  \[\frac A{0-1}+\frac B{0+1}=\frac{0+2}{0^2-1},\] or  \[-A+B=-2.\] To solve for $B$ we add these two expressions: \[4B=-2,\] so $B=\boxed{-\frac12}$.