A question on minimum and maximum values What are the (possible) minimum and maximum values of the rational algebraic expression
$$\frac{AB}{(A + B)(A - B)},$$
if $A, B \in \mathbb{R}$ with $B < A$?
Thank you!
 A: There is no maximum. By choosing, for example, $B=1$, and $A$ a tiny bit bigger than $1$, we can make our expression larger than any prespecified real number.
There is also no minimum.By choosing $A=-1$, and $B$ a tiny bit less than $-1$, we can make our expression less than any prespecified real number.
A: Either $B$ or $A$ must be non-zero (otherwise the given expression is undefined). If one of them is equal to zero, the whole expression is zero, which is clearly not a max or a min. Thus, we may assume that both $B$ and $A$ are non-zero.
Set $r=B/A$. Then
$$\frac{A^2-B^2}{AB} = \frac{A}{B} - \frac{B}{A} = r - \frac{1}{r}.$$ 
Thus, since $r$ can be any real number (except $0$), our problem reduces to finding the max and min of this expression (or rather, of the reciprocal of this expression). If $r\leq1$, then 
$$r-\frac{1}{r} \leq r-1<r,$$
so since $r$ can be as small as we want, the given expression has no minimum. Similarly, if $r\geq1$, then 
$$r-\frac{1}{r} \geq r-1,$$
so since $r$ can be as large as we want, the given expression has no maximum.
