Indeterminate form in The New Yorker

In Calculus II we teach our students about a variety of indeterminate forms: $\displaystyle\frac{0}{0}$, $\displaystyle\frac{\infty}{\infty}$, $\infty-\infty$, etc. I was reminded of another indeterminate form when reading Malcolm Gladwell’s thought-provoking (negative) review of the book Free: The Future of a Radical Price, by Chris Anderson (editor of Wired). The review appears in The New Yorker (that you can read online for free…).

Gladwell describes premise of the book:

[Free] is essentially an extended elaboration of Stewart Brand’s famous declaration that “information wants to be free.” The digital age, Anderson argues, is exerting an inexorable downward pressure on the prices of all things “made of ideas.” Anderson does not consider this a passing trend. Rather, he seems to think of it as an iron law: “In the digital realm you can try to keep Free at bay with laws and locks, but eventually the force of economic gravity will win.”

The indeterminate form that came to my mind was $0\cdot\infty$. If $\displaystyle\lim_{x\to a}f(x)=0$ and $\displaystyle\lim_{x\to a}g(x)=\infty$, then $\displaystyle\lim_{x\to a}f(x)g(x)$ could be anything; $f(x)g(x)$ could be dominated by $f$ and be driven down to zero, or dominated by $g$ and blow up to infinity, perhaps they will will be evenly matched in their fight and $f(x)g(x)$ will tend to a finite nonzero number, or maybe they will fight eternally with no limit at all.

Why is that? Because of the very principles of Free that Anderson so energetically celebrates. When you let people upload and download as many videos as they want, lots of them will take you up on the offer. That’s the magic of Free psychology: an estimated seventy-five billion videos will be served up by YouTube this year. Although the magic of Free technology means that the cost of serving up each video is “close enough to free to round down,” “close enough to free” multiplied by seventy-five billion is still a very large number. [my emphasis].

I’m terribly over-simplifying the situation with this crude analysis, but here goes. Although it may be the case that $\displaystyle\lim_{x\to \infty} c(x)=0$ (where $c(x)$ is the cost to host each video if $x$ videos have been hosted), Gladwell takes issue with Anderson’s assertion that $\displaystyle\lim_{x\to \infty}(x\cdot c(x))\approx 0$, pointing out that it may be the case that $\displaystyle\lim_{x\to \infty}(x\cdot c(x))=\infty$.

I always find that people misunderstand about this indeterminate form. For example many people thought that the value of $\frac{0}{0}$ itself is indeterminate (can be anything) and not realizing that they are indeterminate only in the limit context.