Take the 2-minute tour ×  If I am given the following graph, how
do I compute the limit of x->2 of f(f(x))?  I tried to "compose"
the limits, but lim x->2 f(x) is 1, but then f is not continuous
at 1, so the limit DNE? Is that right?  enter image description
here  share|improve this question That is correct. –  par Dec 16
'13 at 3:02 add comment  2 Answers  up vote 10 down vote
accepted  No, that is not correct. From the graph, no matter how
$x$ approaches $2$, $f(x)$ will approach $1$ from below.
Therefore, to find the limit of $f(f(x))$, you want the one-
sided limit of $f$ at $1$, which does exist and can be seen on
the graph to be $-1$.  share|improve this answer Great, thanks!
I think that then I can also use this "from above" notion to
solve the same limit as x approaches -1 (and get the answer
being 2). But what about as x approaches 4? Then the
above/below/left/right limits all go to 1, and it's not clear
which one-sided limit to use –  MikeS Dec 16 '13 at 4:25 @MikeS:
Draw a graph of $f(f(x))$ for $x$