Given a function $F$ defined over -infinity to +infinity. The limit of $F$ as $x$ tends to plus infinity is equal to $k$, where $k$ is any constant. (asymptote) If we suppose that $k$ is the greatest $F(x)$ possible, why can't we consider it a maximum?
2026-03-30 20:42:57.1774903377
Why is it wrong to consider the highest asymptotic point as an absolute maximum of the function?
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There is no issue to say a value is maximum if it is achieved at some $ x$. But if that value is not achieved at any $ x$, then it can't be maximum. To be called a maximum, say local maximum or global maximum, there should a neighborhood such that value of $x$, is greater than values of the neighborhood. This is why we define infimum and supremum.