Question

Let [t] denote the greatest integer $\leq t$. If for some
$\lambda \in R -\{0,1\}, \displaystyle\lim _{x \rightarrow 0}\left|\frac{1- x +| x |}{\lambda- x +[ x ]}\right|= L ,$ then $L$ is equal to :

• A. 1
• B. 2
• C. $\frac{1}{2}$
• D. 0

Answer 1

Answer: (B)

$LHL : \displaystyle\lim _{x \rightarrow 0^{-}}\left|\frac{1- x - x }{\lambda- x -1}\right|=\left|\frac{1}{\lambda-1}\right|$
$RHL : \displaystyle\lim _{ x \rightarrow 0^{+}}\left|\frac{1- x + x }{\lambda- x +1}\right|=\left|\frac{1}{\lambda}\right|$
For existence of limit
$LHL = RHL$
$\Rightarrow \,\,\,\, \frac{1}{|\lambda-1|}=\frac{1}{|\lambda|}$
$\Rightarrow \lambda=\frac{1}{2}$
$\therefore \,\,\,\, L=\frac{1}{|\lambda|}=2$