I have a question regarding composition of functions.
The question I am asked is:
$f(x)= \begin{cases} 1, & x \geq 0 \\ -1, & x < 0 \end{cases}$ and $g(x)=-(x-2)^2$
why does $$\lim_{x \to 2} f(g(x)) \neq f(\lim_{x \to 2} g(x)).$$
What I would like to know is do I just state that the limit of $g(x)$ as $x$ approaches $2$ is zero and the limit of $f(x)$ at zero doesn't exist since the right and left hand limits are not equal. Or is the question asking for something else?
Thanks in advance,
Guide:
The right hand side is indeed $f(0)$.
The reasoning for the left hand side is wrong. It indeed exists but it is not equal to the right hand side.
Try to find out what is $f(g(x))$ explicitly, then take its limit. To help you, think when does $g(x) < 0$ and when does $g(x) \geq 0$.