Question

If $\frac{\left( x + \sqrt{x^{2} - 4} \right)}{2}$is a polynomial satisfying $\frac{x}{\left( 1 + x^{2} \right)}$and $\frac{\left( x - \sqrt{x^{2} - 4} \right)}{2}$ then $\left( 1 + \sqrt{x^{2} - 4} \right)$is

• A. 2
• B. 1
• C. -1
• D. None of these

Answer 1

Answer: (A)

2

Answer 2

By considering a general nth degree polynomial and

writing in the expression $f ( x ) \cdot f \left( \frac { 1 } { x } \right) = f ( x ) + f \left( \frac { 1 } { x } \right)$,

One can prove by comparing the coefficients of $x ^ { n } , x ^ { n - 1 } , \ldots \ldots$ constant term that the polynomial satisfying the above expression is either of the form $x ^ { n }$ +1 or -xⁿ +1. But $f ( 2 ) = - 2 ^ { n } + 1 > 1$is not possible as $- 2 ^ { n }$ > o for any n. Hence so, $\lim _ { x \rightarrow 1 } f ( x ) = 1 + 1 = 2$