Mathematics - Limits Question with Solution | TestHub

$f\left(1\right)=\frac{1}{2};f\left(-x\right)= -f\left(x\right);f\left(-1\right)= -\frac{1}{2} ad f\left(x\right)$is zero only at one point
$F\left(x\right)= \underset{-1}{\overset{x}{\int }}f\left(t\right)dt$=$\underset{1}{\overset{x}{\int }}f\left(t\right)dtx\in [-1, 2]$as it is an odd function,
$F^{\prime }\left(x\right)=f\left(x\right)$
$G\left(x\right)= \underset{-1}{\overset{x}{\int }}t\left|f\left(f\left(t\right)\right)\right| dt$=$\underset{1}{\overset{x}{\int }}t\left|f\left(f\left(t\right)\right)\right| dt$, as it is an odd function,

$\Rightarrow G^{\prime }\left(x\right)=x \left|f\left(f\left(x\right)\right)\right|$
$\underset{x\to 1}{\lim }\frac{F\left(x\right)}{G\left(x\right)}= \underset{x\to 1}{\lim }\frac{F\prime \left(x\right)}{G\prime \left(x\right)}= \underset{x\to 1}{\lim }\frac{f\left(x\right)}{x\left|f\left(f\left(x\right)\right)\right|}$[ As the limit is in the form of 0/0]
$= \frac{\frac{1}{2}}{\left|f\left(\frac{1}{2}\right)\right|}=\frac{1}{14}$
$\Rightarrow \left|f\left(\frac{1}{2}\right)\right|=7$
$\Rightarrow f\left(\frac{1}{2}\right)=7 as f\left(\frac{1}{2}\right)$can not be negative