Question

For each $t\in R,$ let $\left[t\right]$ be the greatest integer less than or equal to $t.$ Then $\underset{x\to 0^{+}}{lim}x(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+\cdots +\left[\frac{15}{x}\right])$

• A. does not exist (in $R$)
• B. is equal to $0$
• C. is equal to $15$
• D. is equal to $120$

Answer 1

The correct option is D is equal to $120$

Given: $\underset{x\to 0^{+}}{lim}x(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+\cdots +\left[\frac{15}{x}\right])$
$\Rightarrow \underset{x\to 0^{+}}{lim}x[(\frac{1}{x}+\frac{2}{x}+\cdots +\frac{15}{x})-(\left\{\frac{1}{x}\right\}+\left\{\frac{2}{x}\right\}+\cdots +\left\{\frac{15}{x}\right\})]$

$\Rightarrow \underset{x\to 0^{+}}{lim}(1+2+\cdots +15)-x(\left\{\frac{1}{x}\right\}+\cdots +\left\{\frac{15}{x}\right\})$
$=120$