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Evaluate the limit as x approaches 0 of sin([x−3]) divided by [x−3], where [.] denotes the greatest integer function.

1. 0
2. 1
3. the limit does not exist
4. sin 1

Correct answer: the limit does not exist

Solution

Near x=0, x-3 is near -3: for x->0- we get [x-3] = -4 (limit sin4/4) and for x->0+ we get [x-3] = -3 (limit sin3/3). The one-sided limits differ, so the limit does not exist.

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