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

Eric Wofsey
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Nothing
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  • What do you mean by $a(\infty)$? What are $a$ and $b$? – Eric Wofsey Sep 27 '20 at 19:42
  • By the definition of $\infty$, it is "proper" to write $\infty> \frac{b}{a}$ for any a and b as long as $\frac{b}{a}$ exists! It is proper to write $\infty> x$ for any real number x. It is NOT, however, proper to write "$a(\infty)$", meaning "a times $\infty$ since $\infty$ is not a real number and that multiplication is not defined. – user247327 Sep 27 '20 at 19:54
  • at first one might think of plugging in $\infty$ into $a$ like $a$ is the chart in the riemann surface $C_{\infty}$ containing $\infty$. why not preface in saying that $a$ and $b$ are real numbers or something? – BCLC Oct 31 '20 at 06:43

1 Answers1

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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".

Mark S.
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