- you cannot plug infinity into an equation
- something cannot be "equal to infinity"
- "f(x) when x approaches infinity" just means whatever happens to f(x) when you scroll really far to the right on its graph
- "f(x) approaches infinity" just means it increases without bound
- "but what about \(\frac{1}{\infty}=0\)? isn't that a true statement?" it's an informal statement that is informally true. if i say "there isnt nothing you can do about it", you know informally that what i mean is "there isnt anything you can do about it". but if you start to think about the sentence formally, that meaning gets lost.
- this page will review and explain the informal rules of math involving infinity. hopefully this page will help prove to you that infinity is not a thing, but a behavior. this does not make it any less useful in calculus or elsewhere — rather, the moral of the story here is to tread carefully with it.
\[(A)(\infty)\left\{\begin{array}{cl}\infty\text{ if }A>0\\-\infty\text{ if } A < 0\end{array}\right\}\]
below is the graph of \((A)(+\infty)\) for a randomly-chosen positive A. youll notice that the function is increasing the whole time, including on the right side of the graph.
now let's scroll to the right....
to the right some more....
and some more....
at some point, you're forced to come to terms with the fact that this is an incredibly boring graph of a function that will only ever increase. it'll never reach some magic value that'll make it stop increasing; there's no point you can scroll to that'll turn y into infinity and finally break it free of this mortal plane. it just keeps increasing. this behavior is what we refer to when we say \((A)(+\infty)\)=+\infty\) for A>0.