Mathematics - Limits Question with Solution | TestHub

Let, $y=\underset{x\to 0}{\lim }\frac{x{\int }_0^{x}f\left(t\right)dt}{e^{x^2}-1}$

$\Rightarrow y=\underset{x\to 0}{\lim }\frac{x{\int }_0^{x}f\left(t\right)dt}{\left(\frac{e^{x^2}-1}{x^2}\right)\times x^2}$

We know that, $\underset{x\to 0}{\lim }\frac{e^{x^2}-1}{x^2}=1$

$\Rightarrow y=\underset{x\to 0}{\lim }\frac{{\int }_0^{x}f\left(t\right)dt}{x}$

Applying L-hospital's rule and Newton Leibnitz Theorem we get,

$\Rightarrow y=\underset{x\to 0}{\lim }\frac{f\left(x\right)}{1}$

$\Rightarrow y=f\left(0\right)$

It is given that, $f\left(0\right)=\frac{1}{2}$

$\Rightarrow y=\frac{1}{2}$

$\Rightarrow \alpha =\frac{1}{2}$

$\Rightarrow 8{\alpha }^2=2$