Can limit come into modulus of a given function?

Question

Given an extended real valued function $f:R\to \overline{R}$

Is the following true?

$$ \lim_{x \to a} |f(x)| = |\lim_{x \to a}f(x)| $$

Answer 1

Suppose that $f(x) = 1$ for $x \le a$ and $f(x) = -1$ for $x > a$. Then, $\lim_{x \to a} |f(x)| = 1$, but $|\lim_{x \to a} f(x)|$ does not exist since the left end right limits differ, i.e. $$\lim_{x \to a^-} f(x) = 1 \neq -1 = \lim_{x \to a^+} f(x).$$ The claim is true if $\lim_{x \to a} f(x)$ exists.

Answer 2

Sometimes the left can exist even if the right does not. Consider the function that is $-1$ at rational numbers and $1$ at irrational numbers.