Question

$\underset{x\to 0}{lim}\frac{\sum _{r=1}^{100}\left[x^r\right]}{1+|x|}=$

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

Answer 1

The correct option is D does not exist

$\underset{x\to 0}{lim}\frac{\sum _{r=1}^{100}\left[x^r\right]}{1+|x|}$
$\text{L. H. L.}=\underset{x\to 0^{-}}{lim}\frac{\left[x\right]+\left[x^2\right]+\left[x^3\right]+...+\left[x^{100}\right]}{1-x}$

$=\frac{-1+0-1+0-1+...+0}{1-0}=-50$

$\text{R.H.L.}=\underset{x\to 0^{+}}{lim}\frac{\left[x\right]+\left[x^2\right]+\left[x^3\right]+...+\left[x^{100}\right]}{1+x}$

$=\frac{0+0+...+0}{1+0}=0$

$\text{L.H.L.}\ne \text{R. H. L.}$
$\Rightarrow$ Limit does not exist.