Limits of composite functions

[chilly delta]

Hi guys, I'm having trouble with limits of composite functions.

my process (which gave me the right answer for 5-7 (e): evaluate the limit of the inner function first. Limit as x approaches -2 = 1. Then, use 1 to find limit of outer function. Limit as x approaches 1 is -1, but the answer key is -2 and I don’t understand how that is the answer.

[pseudo skiff]

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[chilly delta]

sorry, forgot to mention that I'm trying to solve (f)

[broken basalt]

Note that around x = -2 f(x) is just equal to 2. So, for any value close to x = -2 f(f(x)) will just be equal to f(f(-2)) = f(1) = -2. Thus, the limit is also -2.

This is a tricky example. I recommend trying to graph f(f(x)) first, you'll see it better that way.

In other words, let g(x) = f(f(x)). Then:
g(-2-) = f(f(-2-)) = f(1) = -2
g(-2+) = f(f(-2+)) = f(1) = -2
So, thd limit is -2.

Notice how f(-2-) and f(-2+) are exactly 1, not 1+ or 1-.

[chilly delta]

so if I'm understand correctly, when determining limits of f(f(x)) functions from a graph, if the limit of the inner function is exactly a certain value (like around x=-2, the limit is exactly 1), then we would plug in the resulting limit value into the outer function? so the limit of f(f(x)) would equal -2 and not -1? sorry a bit hard to explain lol

[broken basalt]

Yeah, pretty much.

[chilly delta]

ok thank you

+close