I often heard such a statement "limit of a continuous function is a function of a limit". Is this statement true ? The statement is "$\lim\limits_{x\to a} f(g(x)) = f(\lim\limits_{x\to a} g(x))$ if $f(x)$ is continuous and if $g(x)$ has limit at a". I thought about the following proof.
1) Suppose $\lim\limits_{x\to a} g(x) = b$
2) Now we will do a substitution u=g(x). So as $x$ approaches $a$, $u$ approaches $b$ (from the (1)). Then $\lim\limits_{x\to a} f(g(x)) = \lim\limits_{u\to b} f(u)= f(b)$ (we can write last equality because $f(x)$ is continuous).
3) $f(\lim\limits_{x\to a} g(x))=f(b)$
4) $\lim\limits_{x\to a} f(g(x)) = f(b)= f(\lim\limits_{x\to a} g(x))$
Q.E.D
Is my proof valid ? If not, what will be the proof if this statement is true ? The only thing I don't like in my proof is substitution in (2), is it valid, and if so, is there some formal basis for it ?
The statement $\lim_{g(x)\to b}$ is not defined.
The easiest way to do this is to use the sequential characterization of limits and continuity.
Let $L = \lim_{x \to a}g(x)$ and let $x_n \to a$. By sequential characterization of limits, $g(x_n) \to L$. Then by sequential continuity, $$f(g(x_n)) \to f(L)$$.
Since this holds for any sequence, we have by the sequential characterization of limits that
$$\lim_{x \to a} f(x) = f(L) = f(\lim_{x \to a}g(x))$$
EDIT-
In response to your comment, what is really being said in #2 is that
$$\lim_{x \to a}(f \circ g)(x) = \lim_{g(x) \to b}(f \circ g)(x)$$ which notationally makes no sense.