Question

If $\alpha$ is the positive root of the equation, $p ( x )= x ^{2}- x -2=0$, then $\displaystyle\lim _{x \rightarrow \alpha^{+}} \frac{\sqrt{1-\cos ( p ( x ))}}{ x +\alpha-4}$ is equal to

• A. $\frac{3}{\sqrt{2}}$
• B. $\frac{3}{2}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{1}{2}$

Answer 1

Answer: (A)

$x^{2}-x-2=0$
roots are $2$ & $-1$
$\Rightarrow \displaystyle\lim _{x \rightarrow 2^{+}} \frac{\sqrt{1-\cos \left(x^{2}-x-2\right)}}{(x-2)}$
$=\displaystyle\lim _{x \rightarrow 2^{+}} \frac{\sqrt{2 \sin ^{2} \frac{\left(x^{2}-x-2\right)}{2}}}{(x-2)}$
$=\displaystyle\lim _{x \rightarrow 2^{+}} \frac{\sqrt{2} \sin \left(\frac{(x-2)(x+1)}{2}\right)}{(x-2)}$
$=\frac{3}{\sqrt{2}}$