I have such an expression and I'm trying to find a supremum and infimum of this set:
$A = ${$\frac{70k + 11m}{2k + 13m} : k,m \in N, k \ge 1000, k+m \ge 2000$}
I convert this fraction to: $35 - \frac{444m}{2k+13m}$. Now I don't really know how to estimate this expresson, and take some good numbers to be infimum and supremum. Could you help me and give some explanaiton?
If, for fixed $m$, you let $k \to +\infty$, you see that the supremum is $35$. On the other hand, since $$ \frac{444m}{2k+13m} \leq \frac{444m}{2000+13m} $$ and the last sequence is increasing and converging to $\frac{444}{13}$ (as $m\to \infty$), we see that the expression can take any value as close to $35-\frac{444}{13}$ as we wish, without ever reaching it, which means that the infimum is $\frac{11}{13}$.