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Q. $\displaystyle\lim _{x \rightarrow 1}\left(\frac{\int\limits_{0}^{(x-1)^{2}} t \cos \left(t^{2}\right) d t}{(x-1) \sin (x-1)}\right)$

JEE MainJEE Main 2020Integrals

Solution:

$\displaystyle\lim _{x \rightarrow 1} \frac{\int\limits_{0}^{(x-1)^{2}} t cos (t^2) dt}{(x-1) \sin (x-1)}\left(\frac{0}{0}\right)$
Apply L Hopital Rule
$=\displaystyle\lim _{x \rightarrow 1} \frac{2(x-1) \cdot(x-1)^{2} \cos (x-1)^{4}-0}{(x-1) \cdot \cos (x-1)+\sin (x-1)}\left(\frac{0}{0}\right)$
$=\displaystyle\lim _{x \rightarrow 1} \frac{2(x-1)^{3} \cdot \cos (x-1)^{4}}{(x-1)\left[\cos (x-1)+\frac{\sin (x-1)}{(x-1)}\right]}$
$=\displaystyle\lim _{x \rightarrow 1} \frac{2(x-1)^{2} \cos (x-1)^{4}}{\left.\cos (x-1)+\frac{\sin (x-1)}{(x-1)}\right]}$
$=\displaystyle\lim _{x \rightarrow 1} \frac{2(x-1)^{2} \cos (x-1)^{4}}{\cos (x-1)+\frac{\sin (x-1)}{(x-1)}}$
on taking limit
$=\frac{0}{1+1}=0$