A suitable word for explaining $\lim_{x \to c}f(x)= -\infty$

Question

I am teaching calculus. I teach it in English and, maybe since I am not a native speaker, I stumbled in the following linguistic issue.

We were studying the behavior at 0 of $f(x)=\frac{1}{\sin(x)}$. We first established $\lim_{x \to 0^+}\frac{1}{\sin(x)}=+\infty$. Since some students were confused, I said that "Since we are dividing by a smaller and smaller positive number, we are getting a bigger and bigger number". After this intuitive explanation, everybody seemed satisfied.

Then, we moved to studying $\lim_{x \to 0^-}f(x)$. Students got that this time "the opposite thing was happening", since we were dividing by negative numbers.

Here comes the issue. Since we were doing something of an "opposite flavor", and since the opposite of big is small, some students guessed the limit should be 0.

What I tried to convey them is that "dividing by a smaller and smaller negative number provided us with a more and more negative number". Still, I am not very satisfied with this phrasing, since the temptation of thinking "small" is big, given that the alternative is spelling out "more and more negative".

Question

In your experience, how did you get around this issue? What is a convenient way to phrase what is happening?

Answer 1

In English-Math, the word "small" can be ambiguous. Both "$-100$ is smaller than $-2$" and "$-0.001$ is smaller than $-2$" can be true statements. Perhaps if you explained to your students that by "smaller" you meant "to the left on the number line" and that $-\infty$ is the smallest (in this sense) possible answer, it would clear things up.

I think in the case of "smaller " being "to the left of", I would call it "order smallness" and in the case of it being "closer to zero", I would call it "absolute smallness."

I don't think there is a really super good solution here. Amongst native-English-speaking mathematicians, this often needs to be clarified.

Answer 2

Yes this ambiguity of "small" comes around, I believe, because in our early Math experience with positive numbers big and small works. We are actually describing the size not the order on a number line. When talking about the behaviour of, say, a cubic graph we sometimes talk about "when $x$ gets large and negative" when describing why the graph goes very negative the further left you go on the graph.

If you stick with the terminology "greater" and "less" there is no ambiguity. Reserve big and small for absolute values; $-\infty$ is that which is less than any real number, but in most senses $-\infty$ is very very big! When teaching I clarify this and try to be consistent in my own usage.

In line with your question, when you say bigger in the first case you really are saying more and more positive; if you were testing values one by one the next number will be greater than the last; the opposite of more positive is either less positive or more negative, so it makes perfect sense to talk of the other limit becoming more and more negative.