how do you read: "$\lim (n-1)/(n-2) = 1$"

Question

The limit of the fraction $(n-1)/(n-2)$ with $n$ approaching $+\infty$ is $1$. But how do you read that, concisely? (say during chalking the formula on a blackboard). Is it acceptable to say "lim", for instance?

I would read "the limit of n minus one over n minus two, with n approaching positive infinity is one".

Would that be correct? But then, how do you deal with the ambiguity regarding parentheses? Would you have to say "the limit of the quantity n minus one over the quantity n minus two.." or do you say "left parenthesis n minus one right parenthesis" ?

Answer 1

I tend to read $\lim\limits_{x\to a}f(x)$ as

'The limit as $x$ approaches $a$ of the function $f(x)$ is . . .'

As for dealing with the ambiguity when verbalising a quotient like this, I find that the use of the word 'all' is helpful to distinguish between the possible numerators: $n - 1$ and $1$. I would say

'$n$ minus one all divided by $n$ minus two'

There is still ambiguity as it is not clear whether $n - 1$ is all divided by $n$ or $n - 2$. This is usually avoided by the pacing of the sentence (I'd say '$n$ minus two' faster so that it seems like one object). If I wanted instead to refer to $n - 1$ all divided by $n$ and then minus two from that, I'd change the pacing, putting a little bit of a pause between the final '$n$' and the 'minus two'. You could avoid the ambiguity in this case by instead writing the expression as $-2 + \frac{n-1}{n}$.

Answer 2

I'd always use the full word "limit," but I'd shorten some other terms when I'm writing it on a board. I'd perhaps read it as

The limit of the quotient as $n$ tends to infinity

or to be slightly more wordy,

The limit of $n - 1$ (brief pause) over $n - 2$ as $n$ tends to infinity

But especially if I'm writing it on a board, I'd rather just point at the fraction than have ambiguity in the English.

Answer 3

I would say "The limit as $n$ approaches infinity of $n$ minus one all over $n$ minus two equals one".

Answer 4

Well, if you don't mind a mild loss of grammatical correctness, the way I learnt to say this was, "Limit n (tends to)/(approaches) infinity n-1 whole divided by n-2 equals 1" Sometimes I've noticed people (including myself) even omitting the word "divided"