Question:
If $\alpha$ is the positive root of the equation, $p(x)=x^{2}-x-2=0$, then $\lim _{x \rightarrow a^{+}} \frac{\sqrt{1-\cos (p(x))}}{x+\alpha-4}$
is equal to
Correct Option: 1
Solution:
$x^{2}-x-2=0$
roots are $2 \&-1$
$\Rightarrow \lim _{x \rightarrow 2^{+}} \frac{\sqrt{1-\cos \left(x^{2}-x-2\right)}}{(x-2)}$
$=\lim _{x \rightarrow 2^{+}} \frac{\sqrt{2 \sin ^{2} \frac{\left(x^{2}-x-2\right)}{2}}}{(x-2)}$
$=\lim _{x \rightarrow 2^{+}} \frac{\sqrt{2} \sin \left(\frac{(x-2)(x+1)}{2}\right)}{(x-2)}$
$=\frac{3}{\sqrt{2}}$