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For every real number t, let [t] denote the greatest integer less than or equal to t. Then evaluate lim_(x→ 0⁺) x ([1/x]+[2/x]+⋯+[15/x]).

1. It is 15.
2. It is 120.
3. It does not exist in R.
4. It is 0.

Correct answer: It is 120.

Solution

As x approaches 0 from the positive side, each term [k/x] for k = 1 to 15 approaches k/x, leading to the sum [1/x] + [2/x] +... + [15/x] approximating (1 + 2 +... + 15)(1/x) = 120/x. Multiplying by x gives the limit as 120, confirming that the correct answer is 120.

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