I apologize if this is a stupid question, but I don't really understand this part of limits. I know that for all functions $f(x)$ and $g(x)$ and all numbers $a$,
$$\lim_{x \to a} f(x)g(x) = \lim_{x \to a} f(x) \lim_{x \to a} g(x).$$
I'm not sure if we can substitute other limits into the limits for the individual function, though. For example, say
$$\lim_{x \to a} f(x) = 0$$
and
$$\lim_{x \to a} g(x) = \infty.$$
The limit is obviously of indeterminate form. But what if we consider the limit of a constant multiple of $f(x)$? Wouldn't that be $0$ also?
$$\lim_{x \to a} cf(x) = \lim_{x \to a} f(x)$$
But in that case, can't we substitute the first limit in for the second in the limit of the products so that
$$\lim_{x \to a} f(x)g(x) = \lim_{x \to a} cf(x)\lim_{x \to a} g(x) = c\lim_{x \to a} f(x)\lim_{x \to a} g(x).$$
What am I doing wrong here? Thank you!