$$S=\left\{\frac{5mn-m-4n-1}{2mn-n+2m-1}: m,n \in \Bbb N \right\}$$
I can't find a way to group this fraction into something simpler. The denominator can be written as $(2m-1)(n+1)$ but I don't know what to do with numerator.
2026-03-25 05:41:58.1774417318
Find infimum and supremum of $S=\left\{\frac{5mn-m-4n-1}{2mn-n+2m-1}\right\}$
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Divide the two polynomials.
either with respect to $n:\quad\dfrac{5mn-m-4n-1}{2mn-n+2m-1}=\dfrac{5m-4}{2m-1}-\dfrac{3}{n+1}$
or with respect to $m:\quad\dfrac{5n-1}{2n+2}-\dfrac{3}{4m-2}$
Now the variables are separated and this is easier to study.