Question

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

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

Answer 1

The correct option is A is equal to $120$

$L=\underset{x\to 0+}{lim}x[\left(\frac{1}{x}\right)+...+\left(\frac{15}{x}\right)]$

$L=\underset{x\to 0+}{lim}x\left(\frac{1}{x}\right)+x\left(\frac{2}{x}\right)+....+x\left(\frac{15}{x}\right)$

Take $\underset{x\to 0+}{lim}x\left[\frac{1}{x}\right]$

let $\frac{1}{x}=y$

when $x\to 0$ ,$y\to \infty$

$L=\underset{y\to \infty }{lim}\frac{1}{y}y$

Let $y=n+p$, when $n\to \infty$ , $0<p<1$

$\underset{n\to \infty }{lim}\frac{n}{n+p}$

$\underset{n\to \infty }{lim}\frac{1}{1+\frac{p}{n}}=1$

Take $\underset{x\to 0+}{lim}x\left[\frac{2}{x}\right]$

Let $\frac{2}{x}=y$,

when $x\to 0$,$y\to \infty$,

$\underset{y\to \infty }{lim}\frac{2}{y}y=2$

$\underset{x\to 0+}{lim}x\left[\frac{1}{x}\right]+x\left[\frac{2}{x}\right]+----+x\left[\frac{15}{x}\right]$

$=1+2+3+----+15$

$=\frac{n(n+1)}{2}$

$=\frac{15\times 16}{2}$

$=120$