Question

If α is the positive root of the equation, p(x) = x² – x – 2 = 0, then lim_x→α+(√1-cos(p(x)))/(x + α - 4) is equal to 

(1) 3/√2

(2) 3/2 

(3) 1/√2 

(4) 1/2

Answer 1

Correct option is (1) 3/√2

x² - x - 2 = 0

roots are 2 & -1

⇒ \(\lim\limits_{x\to 2^+} \frac{\sqrt{1 - cos(x^2 - x-2)}}{(x-2)}\)

\(= \lim\limits_{x\to 2^+}\frac{\sqrt{2sin^2\frac{(x^2 - x - 2)}2}}{(x - 2)}\)

\(= \lim\limits_{x \to 2^+} \cfrac{\sqrt 2\sin \left(\frac{(x-2)(x +1)}2\right)}{\frac{(x-2)(x +1)}2 \times \frac 2{x + 1}}\)

\(= \lim\limits_{x \to 2^+} \sqrt 2 \times \frac{x +1}2\)     \(\left(\because \lim\limits_{x\to 2^+} \frac{\sin\left(\frac{(x - 2)(x+1)}2\right)}{\frac{(x -2)(x + 1)}2} = 1\right)\)

\(= \lim\limits_{x\to 2^+} \frac {x + 1}{\sqrt 2}\)

\(= \frac {2 +1}{\sqrt 2}\)

\(= \frac 3{\sqrt 2}\)

Answer 2

The correct option is (1) 3/√2.