If α is the positive root of the equation,

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

  1. $\frac{3}{\sqrt{2}}$

  2. $\frac{3}{2}$

  3. $\frac{1}{\sqrt{2}}$

  4. $\frac{1}{2}$


Correct Option: 1

Solution:

$x^{2}-x-2=0$

roots are $2 \&-1$

$\Rightarrow \lim _{x \rightarrow 2^{+}} \frac{\sqrt{1-\cos \left(x^{2}-x-2\right)}}{(x-2)}$

$=\lim _{x \rightarrow 2^{+}} \frac{\sqrt{2 \sin ^{2} \frac{\left(x^{2}-x-2\right)}{2}}}{(x-2)}$

$=\lim _{x \rightarrow 2^{+}} \frac{\sqrt{2} \sin \left(\frac{(x-2)(x+1)}{2}\right)}{(x-2)}$

$=\frac{3}{\sqrt{2}}$

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