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