$$\lim_{x\rightarrow a} f(x)=L$$
Which way should students best get in the habit of?
According to UCAR-10101: Handbook for Spoken Mathematics, Lawrence A. Chang, Ph.D., page 38, a source for how to speak mathematics to sight-impaired students, we have:
$$\lim_{x\to a} y = b$$ is spoken as the "limit as $x$ approaches $a$ of $y$ equals $b$". For the given expression, $$\lim_{x\to a} f(x) = L$$ is spoken thusly: the "limit as $x$ approaches $a$ of $f$ of $x$ equals $L$."
This is consistent with option $(3)$.
Alternatively, in analysis, from ANALYSIS TAUGHT BY BJORN POONEN, NOTES BY SANATH DEVALAPURKAR, we have the following:
We write $$\lim_{x→a} f(x) =L$$ to say that for every $\epsilon >0,$ $f$ is eventually within $\epsilon$ of $L$ as $x$ approaches $a$”. This means that there exists $δ >0$ such that $0<|x−a|< δ,$ then $|f(x)−L|< \epsilon$.
ADDED: Please see this mathematics.se post about "speaking mathematics". There are additional resources for those interested.
I say it the third way, for these reasons:
Firstly, from a notation point of view, the “$x\to a$” has to be written with the “$\lim$”, and no “$\lim$” can be written without it (without specifically saying what you mean by not having it), so it makes sense to put them together when you say it aloud.
Indeed, you could argue that $\displaystyle\lim_{x \to a}$ is an operation you perform on the function $f(x).$ That is, you can change what $x$ the limit is being found at and also what function you are doing it to.
Secondly the “$\lim_{x \to a} f(x)$” is a thing all by itself, and so it makes sense to say it all together.
As $x$ approaches $a$, $f(x)$ approaches $L$.
First, we emphasize what is happening to the independent variable, then we explain the consequence. I think that this phrasing is concise and easy to understand. It is clean and efficient. This is essentially (3), but I think that the sub-clause "The limit..." is unnecessary.
Moreover, if we have a fixed function and want to consider limits at several points, it provides a consistent framework. For example, consider the rational function $$ f : \mathbb{R}\setminus\{\pm 1\} \to \mathbb{R} : x \mapsto \frac{x+1}{x^2 - 1}. $$ As $x$ approaches $\pm \infty$, $f(x)$ approaches $0$. On the other hand, as $x$ approaches $-1$, $f(x)$ approaches $1/2$, and as $x$ approaches $1$, $f(x)$ is unbounded (in either the positive or negative direction, depending on whether the limit is taken from the left or the right).
I usually say
f(x) lähestyy L:ää, kun x lähestyy a:ta
and sometimes I instead say it more colloquially as
f(x):n raja on L, kun x on a.
The inverted
Kun x lähestyy a:ta, f(x) lähenee L:ää
is also fine and in use.
These would correspond to 2) and 4) in English. I would avoid linguistic complexity, such as a side clause embedded in the sentence, since it might reduce clarity.
You are interchangeable, like peas in a pod.
Cauchy and Weierstrass were usually saying "$f(x)$ becomes arbitrarily close to $L$", with the qualifier "as $x$ approaches $a$", sometimes before, sometimes after, sometimes implied.
They also followed Leibnitz and Lagrange to talked about a quantity $f(x)$ becoming infinely close to $L$, when $|x-a|$ is infinitely small or an infinitesimal.
Either 1 or 3, for the reasons given in the other answers. But I would also note that "as $x$ approaches $a$" need not be set aside by commas (or a break in speech), as it is not optional in the sentence.
Besides the arrangement of the words, it is worth noting that ISO 80000-2, Mathematical signs and symbols to be used in the natural sciences and technology (2009), lists "$x$ tends to $a$" as the verbal equivalent for the notation $x \rightarrow a$, in contrast to "$x$ approaches $a$."
1 > 3 > 2
I prefer the first wording. You get more of the idea of thinking about what x causes what y. Yes, the second is equivalent, but it is awkward, to add the condition (approaching a) after the result. Three is OK also. Although I mildly prefer 1. Maybe shorter.
