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If the limit \(\lim_{x\to ?}\big(-x+c\lfloor x-1\rfloor+d\lfloor 1+x\rfloor\big)\) exists, where \(\lfloor\,\cdot\,\rfloor\) denotes the floor function, what are the possible values of \(c\) and \(d\)?

1. \(c=\frac13, d=1\)
2. \(c=1, d=-1\)
3. \(c=9, d=-9\)
4. \(c=2, d=\frac23\)

Correct answer: \(c=1, d=-1\)

Solution

A limit involving floor functions exists only if the discontinuous jumps from the floor terms cancel on both sides of the point of approach. Matching the left-hand and right-hand limits gives a linear condition on \(c\) and \(d\), which is satisfied by \(c=1\) and \(d=-1\).

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