Question

If g(x) is a polynomial satisfying $g\left(x\right)g\left(y\right)=g\left(x\right)+g\left(y\right)+g\left(xy\right)-2$ for all real x and y and $g\left(2\right)=5$ then $\underset{x\to 3}{lim}g\left(x\right)$ is

• A. 9
• B. 25
• C. 10
• D. none of these

Answer 1

The correct option is C 10

Putting $x=2$ and $y=1$, we have
$g\left(2\right)g\left(1\right)=g\left(2\right)+g\left(1\right)+g\left(2\right)-2$

$\Rightarrow 5g\left(1\right)=8+g\left(1\right)\Rightarrow g\left(1\right)=2$.
Putting $y=1/x$ in the given equation, we have
$g\left(x\right)g\left(1/x\right)=g\left(x\right)+g\left(1/x\right)+g\left(1\right)-2$

$=g\left(x\right)+g\left(1/x\right)$.
Hence as $g\left(x\right)=x^n+1$ or $g\left(x\right)=-x^n+1$. But $5=-2^n+1$ so $-2^n=4$ which is not possible.

Thus $5=g\left(2\right)=2^n+1$
$\Rightarrow n=2.$ Hence, $\underset{x\to 3}{lim}g\left(x\right)=\underset{x\to 3}{lim}(x^2+1)=10$

Hence, option 'C' is correct.