Let [t] denote the greatest integer [ ] t.

Question

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

$$\lambda $$ $$ \in $$ R - {1, 0}, $$\mathop {\lim }\limits_{x \to 0} \left| {{{1 - x + \left| x \right|} \over {\lambda  - x + \left[ x \right]}}} \right|$$ = L, then L is
equal to :

Accepted Answer

Here $$\mathop {\lim }\limits_{x \to 0} \left| {{{1 - x + \left| x \right|} \over {\lambda  - x + [x]}}} \right| = L$$

Here L.H.L. $$\mathop {\lim }\limits_{h \to 0^-} \left| {{{1 + h + h} \over {\lambda  + h - 1}}} \right| = \left| {{1 \over {\lambda  - 1}}} \right|$$

R.H.L. = $$\mathop {\lim }\limits_{h \to 0^+} \left| {{{1 - h + h} \over {\lambda  + h + 0}}} \right| = \left| {{1 \over \lambda }} \right|$$

$$ \because $$ Limit exists. Hence L.H.L. = R.H.L.

$$ \Rightarrow $$ $$\left| {\lambda  - 1} \right| = \left| \lambda  \right|$$

$$ \Rightarrow $$ $$\lambda  = {1 \over 2}$$

$$ \therefore $$ L = $${1 \over {\left| \lambda  \right|}}$$ = 2

Solutions · 1