Limit of f(f(x)) at x=-2: Does it Exist?

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The discussion centers on the limit of the function f(f(x)) as x approaches -2. It is established that the limit exists if the function f(x) is defined and continuous at that point. For example, if f(x) = x, the limit clearly exists. The conversation also touches on the behavior of limits approaching from different sides, emphasizing the need for a specific function definition to determine limit existence accurately.

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Consider the following
lim f(f(x))
x->-2

When looking at the graph of a function, if the function approaches -2 from above on the right
side, and approaches -2 from below on the left side, does the limit exist? I've been told it doesn't, but I don't understand why.
 
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You need to specify what your function is.
If f(x)=x, for example, the limit certainly exists.
 
Only the graph is given.

I guess you could suppose this is a similar graph.
http://curvebank.calstatela.edu/limit/grid3.gif

Then lim f(x) (lets suppose)
x->1

Since the function approaches from above on the right side , and below on the left side, does the limit exist??
 

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