Question

The graph of the function f is shown above. What is limx→−1f(f(x)) ??

Answer 1

Answer: Does not exist

Reason

Limits of function compositions can be tricky. So we have to be careful how we define them. Consider two functions f(x) and g(x). The composite function we'll form is f( g(x) )

Then using limit laws we can state the following:

`\displaystyle\text{If }\lim_{\text{x}\to c} g(\text{x}) = b\text{ and } \lim_{\text{x}\to b} f(\text{x}) = f(b)\text{ then } \lim_{\text{x}\to c} f(g(\text{x})) = f(\lim_{\text{x}\to c} g(\text{x})) = f(b)`

Carefully study those limit equations. Notice how we start with the inner function g(x).

For the composite limit
`\displaystyle\lim_{\text{x}\to -1} f(f(\text{x}))` we first need to find the more simple limit of
`\displaystyle\lim_{\text{x}\to -1}f(\text{x})` which is the inner function.

Approach x = -1 from either left or right, and you should get closer and closer to y = f(x) = 3

Therefore,

`\displaystyle\lim_{\text{x}\to -1} f(\text{x}) = 3`

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Then according to that rule mentioned above, we go from

`\displaystyle\lim_{\text{x}\to -1} f(\text{x}) = 3`

to

`\displaystyle\lim_{\text{x}\to 3} f(\text{x})`

This is where we run into a problem. Approach x = 3 from the left and we get closer to f(x) = -1. But if we approach x = 3 from the right, then f(x) gets closer to f(x) = 1/2 = 0.5

The left hand limit LHL and right hand limit RHL do not agree on the same number.

`\text{LHL} \\e \text{RHL}`

Therefore, the limit
`\displaystyle\lim_{\text{x}\to 3}f(\text{x})` does NOT exist.

This will cause the composite limit
`\displaystyle\lim_{\text{x}\to -1} f(f(\text{x}))` to also NOT exist