Let $ f(x) = x^2$
What is $\displaystyle\lim_{x \to 1}f(3)$
What is this statement saying in plain english?
Is it "What is $f(3)$ approaching as $x$ approaches $1$"?
Let $ f(x) = x^2$
What is $\displaystyle\lim_{x \to 1}f(3)$
What is this statement saying in plain english?
Is it "What is $f(3)$ approaching as $x$ approaches $1$"?
Your interpretation is correct.
As written, it may help to think of this in the following manner: define $g(x) = f(3) = 9$, (i.e. $g$ is a constant function). Then
$\displaystyle\lim_{x \rightarrow 1} f(3) = \displaystyle\lim_{x \rightarrow 1} g(x) = \displaystyle\lim_{x \rightarrow 1} 9.$
Of course the value of the limit is $9$.
I think the "purpose" of this is to explain notation, but that's only a guess.
$f(3) = 9$
So that the rule for limits that applies here is;
Limit of a constant, $b$:
$\lim_{x\rightarrow c} b = b$
where $c=1$ and $b=9$.