TITLE: The limit of $f$ or the limit of $f(x)$?
QUESTION [6 upvotes]: I have read before that $f$ denotes the function $f$ whilst $f(x)$ denotes the value of the function $f$ at $x$. What is right? To say that the limit of $f$ as $x$ tends to $a$ is $L$ or to say that the limit of $f(x)$ as $x$ tends to $a$ is $L$? Put another way, is it the limit of the function at $x$ or the limit of the value of the function at $x$? If both are correct, what is more precise?
Hopefully, you'll understand what I mean.
Thanks in advance.

REPLY [5 votes]: Spivak, in chapter 5 of Calculus, says this:

The equation \begin{equation*} \lim_{x\to a} f(x) = \ell \end{equation*} 
  has exactly the same meaning as the phrase
  \begin{equation*} f \text{ approaches } \ell \text{ near } a.\end{equation*}

He goes on to say:

Notice that our new notation introduces an extra, utterly irrelevant
  letter $x$, which could be replaced by $t$,$y$, or any other letter
  which does not already appear -- the symbols \begin{equation*} \lim_{x\to a} f(x),\quad \lim_{t\to a} f(t), \quad \lim_{y\to a} f(y),\end{equation*} all denote precisely the same number, which depends on
  $f$ and $a$, and has nothing to do with $x,t$, or $y$ (these letters,
  in fact, do not denote anything at all).  A more logical symbol would
  be something like $\lim_a f$, but this notation, despite its brevity,
  is so infuriatingly rigid that almost no one has seriously tried to
  use it.  The notation $\lim_{x \to a} f(x)$ is much more useful
  because a function $f$ often has no simple name, even though it might
  be possible to express $f(x)$ by a simple formula involving $x$.

REPLY [2 votes]: In this case it doesn't much matter -- even in the case that $f$ is a function of multiple variables, etc. it is hard to get too confused.
The statement that $f(x)$ denotes a value in the range of $f$, and not a function, is far from a universal convention. For instance
$$\frac{\partial}{\partial x}f(x,y)$$
is particularly confusing: is it the derivative of $f$ with respect to the first factor, evaluated at $(x,y)$? The derivative of the function $[f(x,y)](x)$ with respect to $x$? What if you saw instead $$\frac{\partial}{\partial x}f(x^2,y)$$
or
$$\frac{\partial}{\partial x}f(\cdot, x)?$$