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Please see my attached image, which is a screenshot from Khan Academy on the limits of composite functions.
I just want to check if I'm understanding this correctly, particularly for #1, which has work shown on the picture.
Now my question:
We are taking the limit of a composition of functions, namely f(x) + g(x).
Now for the limit laws to work, as I understand, a limit has to exist with what you're starting with, and a limit has to exist for what you turn it into. In other words, a limit must exist in the RHS and LHS.
Now for problem 1, the limit of f(x) as x approaches 2 clearly does not exist. And the limit of g(x) as x approaches 2 does not exist. So how can we use the limit laws on this? A limit doesn't exist at the value that we are taking the limit of for each of the functions in the composition?
In the video, he goes on to take the limit of f(x) and g(x) as x approaches 2 from the left, and separately from the right. And these limits each sum to 4, so then it is said that the limit of the composition is 4.
I do understand this reasoning, but what isn't sitting right with me is the fact that the limits didn't exist at x=2 in the first place, but by going at x=2 from each side individually, the limit now exists.
I just want to check if I'm understanding this correctly, particularly for #1, which has work shown on the picture.
Now my question:
We are taking the limit of a composition of functions, namely f(x) + g(x).
Now for the limit laws to work, as I understand, a limit has to exist with what you're starting with, and a limit has to exist for what you turn it into. In other words, a limit must exist in the RHS and LHS.
Now for problem 1, the limit of f(x) as x approaches 2 clearly does not exist. And the limit of g(x) as x approaches 2 does not exist. So how can we use the limit laws on this? A limit doesn't exist at the value that we are taking the limit of for each of the functions in the composition?
In the video, he goes on to take the limit of f(x) and g(x) as x approaches 2 from the left, and separately from the right. And these limits each sum to 4, so then it is said that the limit of the composition is 4.
I do understand this reasoning, but what isn't sitting right with me is the fact that the limits didn't exist at x=2 in the first place, but by going at x=2 from each side individually, the limit now exists.
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