Inequality operations on infinity

Question

if $a(\infty)>b$ and $a>0$, then is it proper to write $\infty>\frac{b}{a}$?

Answer 1

The details depend on the context.

Calculus

In usual calculus, it's not standard practice to write inequalities with $\infty$ like this. And $a(\infty)$ would just be a shorthand for a limit of the form $\lim_{x\to c}f(x)g(x)$ where $\lim_{x\to c}f(x)=a$ and $\lim_{x\to c}g(x)=\infty$ in that $g$ grows without bound as $x$ approaches $c$.

So I would say no, because inequalities likes "$\infty>\frac{b}{a}$ are not meaningful.

Extended reals

Here I will assume $a$ and $b$ are extended reals.

Since $a>0$, $a(\infty)=\infty$ by rules of arithmetic. Then $\infty>b$ for sure.

In the special case where $b=-\infty$ and $a=\infty$, $\frac{b}{a}$ is undefined. Therefore, the answer is no, not in general.

But in all other cases, either $b$ is finite so $\frac{b}{a}$ is defined and finite (so $\infty>\frac{b}{a}$), or $a$ is finite so $\frac{b}{a}$ is defined and not $\infty$ (but possibly $-\infty$) and we still have $\infty>\frac{b}{a}$.

So if you meant for $b$ or $a$ to be finite and your inequalities and arithmetic to be done with the extended reals, then the answer would be "yes".