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Let f(x) = [1 - x*(1 + |1 - x|)] / (|1 - x| * cos(1/(1-x))) for x not equal to 1. Which of the following is/are correct?

1. lim(x -> 1+) f(x) does not exist
2. lim(x -> 1-) f(x) does not exist
3. lim(x -> 1-) f(x) = 0
4. lim(x -> 1+) f(x) = 0

Correct answer: lim(x -> 1-) f(x) does not exist

Solution

From the right (x > 1): the expression simplifies such that the cosine term oscillates infinitely and the limit does not exist. From the left (x < 1): the numerator goes to zero while the denominator also goes to zero, but the cosine oscillation means the limit does not exist.

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