Solve this following


$\lim _{x \rightarrow 1}\left(\frac{\int_{0}^{(x-1)^{2}} t \cos \left(t^{2}\right) d t}{(x-1) \sin (x-1)}\right)$

  1. does not exist

  2. is equal to $\frac{1}{2}$

  3. is equal to 1

  4. is equal to $-\frac{1}{2}$

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


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