Question

$\underset{x\to 0^{-}}{lim}\frac{{\sum }_{r=1}^{2n+1}\left[x^r\right]+(n+1)}{1+\left[x\right]+|x|+2x}$ $($where $nϵN$ & $[$.$]$ denotes the greatest integer function$)$ equals

• A. $-1$
• B. $0$
• C. $1$
• D. Does not exist

Answer 1

Answer: (B)

$0$

$\underset{x\to 0^{-}}{lim}\frac{{\sum }_{r=1}^{2n+1}\left[x^r\right]+(n+1)}{1+\left[x\right]+|x|+2x}=\underset{x\to 0^{-}}{lim}\frac{{\sum }_{r=1}^{2n+1}\left[x^r\right]+(n+1)}{x}$

$\underset{x\to 0^{-}}{lim}\frac{\left[x\right]+\left[x^2\right]+\left[x^3\right]+.....\left[x^{2n+1}\right]+(n+1)}{x}$

$=\underset{x\to 0^{-}}{lim}\frac{\stackrel{\underset{}{(-1)+(-1)+....+(-1)}+(n+1)}{(n+1)times}}{x}=\frac{0}{x}=0$