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EEJ MAIN Mathematics QUESTION #6951
Question 1

For each $t\in\mathbb{R}$, let $[t]$ denote the greatest integer $\leq t$. Evaluate: $\displaystyle\lim_{x\to0^+} x\left(\left[\dfrac{1}{x}\right]+\left[\dfrac{2}{x}\right]+\cdots+\left[\dfrac{15}{x}\right]\right)$

  • Equal to 120✔️
  • Does not exist
  • Equal to 0
  • Equal to 15
Correct Answer Explanation

For large $N=1/x$, $[k/x]\approx kN$ with error at most 1.

$x\sum_{k=1}^{15}\left[\dfrac{k}{x}\right] \approx x\sum_{k=1}^{15}\dfrac{k}{x} = \sum_{k=1}^{15}k = \dfrac{15\cdot16}{2} = 120$

More rigorously: $[k/x] = k/x - \{k/x\}$ where $0\leq\{k/x\}<1$. Then $x\cdot\{k/x\}