Let [t] denote the greatest integer
Question:

Let $[t]$ denote the greatest integer $\leq t$. If for some

$\lambda \in \mathbf{R}-\{0,1\}, \lim _{x \rightarrow 0}\left|\frac{1-x+|x|}{\lambda-x+[x]}\right|=L$, then $L$ is equal to $:$

1. (1) 1

2. (2) 2

3. (3) $\frac{1}{2}$

4. (4) 0

Correct Option: , 2

Solution:

Given $\lim _{x \rightarrow 0}\left|\frac{1-x+|x|}{\lambda-x+[x]}\right|=L$

Here, L.H.L. $=\lim _{h \rightarrow 0}\left|\frac{1+h+h}{\lambda+h-1}\right|=\left|\frac{1}{\lambda-1}\right|$

R.H.L. $=\lim _{h \rightarrow 0}\left|\frac{1-h+h}{\lambda+h+0}\right|=\left|\frac{1}{\lambda}\right|$

Given that limit exists. Hence L.H.L. = R.H.L.

$\Rightarrow|\lambda-1|=|\lambda|$

$\Rightarrow \lambda=\frac{1}{2}$ and $L=\left|\frac{1}{\lambda}\right|=2$