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Q.

For each $t \in R$ ,[t] represents greatest integer less than or equal to t then $\lim_{x \to 0^{+}} x ([\frac{1}{x}] + [\frac{2}{x}] + \dots + [\frac{15}{x}]) =$

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a

15

b

120

c

Does not exists

d

0

answer is B.

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Detailed Solution

= $\lim_{x \to 0^{+}} x (\frac{1 + 2 + 3 + \dots + 15}{x} - (\{\frac{1}{x}\} + \{\frac{2}{x}\} + \dots + \frac{15}{x}\})$
= $\lim_{x \to 0^{+}} \sum 15 - x (\{\frac{1}{x}\} + \{\frac{2}{x}\} + \dots + \{\frac{15}{x}\})$
$\Rightarrow \sum 15 = \frac{15 \times 16}{2} = 120 [0 \leq \{\frac{r}{x}\} < 1 \Rightarrow 0 \leq x \{\frac{r}{x}\} < x]$

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