Finding $\lim f(f(x))$ on a discontinuous function

Question

If you had a piecewise function like the one in the image below, and were asked to find $\lim_{x\to-1} f(f(x))$, how would you go about solving it?

Would you first look at the limit approaching -1, then plug it in? Or would you take the limit of the value of the limit, making the answer undefined?

Thanks in advance

Graph Example

score 3 · Accepted Answer · answered Sep 29 '19 at 19:26

3

Note that as $x$ approach $-1$ by the left then $f(x)$ approach to $2$ from below, so $f(x)$ approach to $2$ from the left, so

$$ \lim_{x\to -1^-}f(f(x))=5 $$

The same happens when you approach to $-1$ by the right, so the limit is $5$.

answered Sep 29 '19 at 19:26

Masacroso

30,417

But how is the limit equal to 5 if the limit at x approaches 2 doesn't exist? – Mace Windu Sep 29 '19 at 19:52
@starwarsmace tha lateral limits approaching to $2$ both exists, but they are different. When we approach to $-1$ we are approaching to $2$ only from below, what is the same to say that we approach to $2$ only from the left – Masacroso Sep 29 '19 at 19:54
So, by that same thinking: https://pasteboard.co/IzHsMIR.png at lim x approaches -1 would be -1? – Mace Windu Sep 29 '19 at 20:25
@starwarsmace exactly – Masacroso Sep 29 '19 at 20:45

score 0 · Answer 2 · answered Sep 29 '19 at 20:04

0

The function is $$f(x) =\begin{cases}x+3 & x<-1 \\0 & x=-1\\x^2+1 & -1<x<2\\-x+4 &x\geq 2\end{cases} $$hope it helps you.

answered Sep 29 '19 at 20:04

Khosrotash

24,922

Finding $\lim f(f(x))$ on a discontinuous function

2 Answers2