Prove the following
Question:

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

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

2. (2) is equal to 0

3. (3) is equal to $-\frac{1}{2}$

4. (4) does not exist

Correct Option: , 2

Solution:

$\lim _{x \rightarrow 1} \frac{\frac{1}{2} \sin (x-1)^{4}}{(x-1) \sin (x-1)}$

Let $x-1=h$ when $x \rightarrow 1$ then $h \rightarrow 0$

$\lim _{h \rightarrow 0} \frac{\sin h^{4}}{h^{4}} \times \frac{h}{\sin h} \times h^{2}=1 \times 1 \times 0=0$