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We know that:

$\lim_{n\to \infty} \frac{\infty}{\infty} = \text{indeterminate}$

And that:

$\lim_{n\to \infty} \frac{n}{n} = 1$

How can I easily explain the difference to first-year university students?

Michael Hardy
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TheNotMe
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    Just explain that $n\ne\infty$, all the more so as $\infty$ is but a metaphor. – Bernard Dec 25 '17 at 10:44
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    The top should read $\frac{\infty}{\infty}=$ indeterminate. It makes no sense to divide infinities as it is not a number; the concept of a limit (which is used in the second one) allows us to treat infinities and the like with more precision. As well as this saying that the limit of a function approaches a value is definitely not the same as saying the function is equal to that value evaluated at that point – aidangallagher4 Dec 25 '17 at 10:46
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    But $\text {lim}{n \to \infty} n = \infty$ is a correct expression (it says that there is no number that is the limit of the sequence ${ n }$) while $\text {lim}{n \to \infty} \infty$ makes no sense. – Mauro ALLEGRANZA Dec 25 '17 at 11:00
  • Related: What is infinity divided by infinity? – user2314737 Aug 27 '20 at 10:24

2 Answers2

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You write

We know that: $\lim_{n\to \infty} \frac{\infty}{\infty} = indeterminate$

but I'd say that we don't know that at all. (Indeed, it makes no sense).

What we do know is that if $$ \lim_{n \to \infty} f(n) = \infty $$ and $$ \lim_{n \to \infty} g(n) = \infty $$ then $$ \lim_{n \to \infty} \frac{f(n)}{g(n)} $$ cannot be directly computed with a quotient rule.

I hate to say this, but I think your problem is not with "explaining this to first-year university students", but rather with knowing and understanding the main theorems yourself before trying to explain them to others.

John Hughes
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  • By "we know that" I indeed meant what you explained in more details. – TheNotMe Dec 25 '17 at 10:51
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    Teaching mathematics well involves (among many other things) saying the true stuff and avoiding the nonsense pretty conscientiously, lest your students start imitating you and thinking that they're saying something true. When you start by asserting nonsense, you're headed down a bad path, IMHO. – John Hughes Dec 25 '17 at 11:01
  • Thank you for your very valuable input. – TheNotMe Dec 25 '17 at 11:26
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    I strongly agree with this answer. $\infty/\infty$ is an advanced shorthand that shouldn’t be shown to first year students. – Stella Biderman Dec 25 '17 at 13:19
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I may first give an example : finding limit $$ \lim_{x \rightarrow \infty} \frac{1+x}{x} $$

When we use straightforward approach, we get $$ \frac{\infty+1}{\infty} = \frac{\infty}{\infty} $$ In the process of investigating a limit, we know that both the numerator and denominator are going to infinity.. but we dont know the behaviour of each dynamics. But if we investigate further we get : $$ 1 + \frac{1}{x} $$ Some other examples :

  1. Numerator might get larger than denomenator exactly $m$ times. The limit will be $m$ : for example $\lim \frac{mx}{x}$. Or the opposite : for example $\lim \frac{x}{mx}$
  2. The numerator gets way too large than denominator : $\lim \frac{x^2}{x}$ , the limit is clearly $\infty$. Or the opposite : $\lim \frac{x}{x^2}$.

Thanks.

Redsbefall
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