Let f:[-( )/(2),( )/(2)] R be a differentiable function such that f(0)=(1)/(2).

Question

Let $f:[-\frac{\pi }{2},\frac{\pi }{2}]	o R$ be a differentiable function such that $f(0)=\frac{1}{2}$. If the $\underset{x	o 0}{lim}\frac{x{\int }_{x}^{0}f(t)dt}{{e}^{{x}^{2}}-1}=\alpha $, then $8{\alpha }^{2}$ is equal to :

Accepted Answer

$\begin{gathered} \underset{x	o 0}{lim}\frac{x{\int }_{x}^{0}f(t)dt}{(\frac{{e}^{{x}^{2}}-1}{{x}^{2}})	imes {x}^{2}} \ \underset{x	o 0}{lim}\frac{{\int }_{x}^{0}f(t)dt}{x}\, (\underset{x	o 0}{lim}\frac{{e}^{{x}^{2}}-1}{{x}^{2}}=1) \ =\underset{x	o 0}{lim}\frac{f(x)}{1} \ f(0)=\frac{1}{2} \ \alpha =\frac{1}{2} \ 8{\alpha }^{2}=2 \end{gathered}$

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