The limit of $f$ or the limit of $f(x)$? 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.
 A: 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 $x$.

A: Either way of speaking is common. You can speak about


*

*The limit of the function $f$ as its argument tends to $42$.


or


*

*The limit of the expression $3x^2+5x+\frac{\log x}{x}$ as $x$ tends to $42$.


In the second case, you can in particular consider the expression $f(x)$.
Neither of these is really more correct or precise than the other one. In mathematics you're generally supposed to pass back and forth effortlessly between a "function" and an "expression with a free variable in it", and use the perspective that makes most sense in any given context.
In the case of limits, there's the peculiarity that the formal definition of limits is usually phrased in terms of functions (because an expression is usually not considered a "thing" in itself that it looks nice to quantify a definition over, and a function is exactly how we pack up an expression as a thing when we need to), whereas the usual notation for concrete limits always works on expressions -- if you write $\lim_{x\to 42} 3x^2+5x+\frac{\log x}{x}$ there's no named function in sight, but the limit is perfectly good nevertheless.
However, don't say "the limit of $f$ for $x\to 42$. If you want to name the independent variable, you need to speak about the limit of an expression rather than a function.
A: By definition one writes $\lim \limits_{x\to a}\left(f(x)\right)=L$, if, and only if, there exists $L\in \Bbb R$ such that  $$\forall \varepsilon>0\exists \delta >0\forall x\in D_f \left(|x-a|<\delta\implies |f(x)-L|<\varepsilon\right).$$
This ($\lim \limits_{x\to a}\left(f(x)\right)=L$) is simply a notation to convey the what is above, it's an abbreviation. In fact it isn't even a proper equality, it's just a string of symbols that happens to have the symbol $=$ in the middle. One only earns the right to define $\lim \limits_{x\to a}\left(f(x)\right)$ as being equal to something after proving that an $L$ on the conditions above, must be unique.
If you want to mention the limit orally, I think it makes more sense to say 'limit of $f$ as $x$ approaches $a$' (with the implied assumption that $f$ is a function on the variable $x$) than 'limit of $f$ of $x$ as $x$ approaches $a$', because, as I said, $\lim \limits_{x\to a}\left(f(x)\right)=L$ is just a notation and it's meaningless by itself, it's just short for what I said above.
A: 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
$$\frac{\partial}{\partial x}f(\cdot, x)?$$
