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

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In summary, the question being asked is whether the limit exists for the function lim f(f(x)) as x approaches -2. The person asking the question is unsure because they have been told it does not exist but they do not understand why. The responder clarifies that the question needs to specify what the function is and provides an example where the limit does exist. The person then provides a link to a graph and asks if the limit exists for this graph.
  • #1
NATURE.M
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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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  • #2
You need to specify what your function is.
If f(x)=x, for example, the limit certainly exists.
 
  • #3
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??
 

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

What is the limit of f(f(x)) as x approaches -2?

The limit of f(f(x)) at x=-2 is the value that f(x) approaches as x gets closer and closer to -2. It is also known as the limit of composition of functions.

How can we determine if the limit of f(f(x)) at x=-2 exists?

We can determine if the limit of f(f(x)) at x=-2 exists by evaluating the function at -2 and checking if the value is finite. If the value is finite, then the limit exists. We can also use the squeeze theorem or the limit laws to determine the existence of the limit.

Can the limit of f(f(x)) at x=-2 be different from the limit of f(x) at x=-2?

Yes, the limit of f(f(x)) at x=-2 can be different from the limit of f(x) at x=-2. This is because the limit of f(f(x)) takes into account the behavior of both f(x) and f(f(x)) as x approaches -2, while the limit of f(x) only considers the behavior of f(x) itself.

What happens if the limit of f(x) at x=-2 does not exist?

If the limit of f(x) at x=-2 does not exist, it is not possible to determine the limit of f(f(x)) at x=-2. This is because the limit of f(f(x)) at x=-2 is dependent on the limit of f(x) at x=-2. If the latter does not exist, the former cannot be determined.

How does the behavior of f(x) affect the limit of f(f(x)) at x=-2?

The behavior of f(x) can affect the limit of f(f(x)) at x=-2 in several ways. If f(x) has a finite limit at x=-2, then the limit of f(f(x)) at x=-2 will also be finite. If f(x) has a vertical asymptote or a jump discontinuity at x=-2, then the limit of f(f(x)) at x=-2 will not exist. Additionally, the limit of f(f(x)) at x=-2 can also be affected by the behavior of f(x) near x=-2, such as if f(x) has a removable discontinuity at x=-2.

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