x ( )/(2)lim((|1(x-( )/(2))^2 _x^3^(( )/(2))^3 (t^(1)/(3))dt))| is equal to

Question

$\underset{x	o \frac{\pi }{2}}{lim}((|\frac{1}{{(x-\frac{\pi }{2})}^{2}}\int _{{x}^{3}}^{{(\frac{\pi }{2})}^{3}}\cos ({t}^{\frac{1}{3}})dt))|$ is equal to

Accepted Answer

Using L'hospital rule

$\begin{gathered} =\underset{x	o \frac{{\pi }^{-}}{2}}{lim}\frac{0-\cos x	imes 3{x}^{2}}{2(x-\frac{\pi }{2})} \ =\underset{x	o \frac{{\pi }^{-}}{2}}{lim}\frac{\sin (x-\frac{\pi }{2})}{2(x-\frac{\pi }{2})}	imes \frac{3{\pi }^{2}}{4} \ =\frac{3{\pi }^{2}}{8} \end{gathered}$

Solutions · 1