If `g (x)` is a polynomial satisfying `g(x)g(y) =g(x) + g(y) + g (xy) -2` for all real `x` and `y` and `g (2) = 5` then `lim_(x->3) g (x)` is

Options

A. 6
B. 25
C. 24
D. 26

Answer

D

Explanation

Putting x=2 and y=1 in the given relation , we obtain
g(2)g(1)=g(2)+g(1)+g(2)-2
`implies 5g(1)=5+g(1)+5-2`
`implies g(1)=2`
Putting `y=(1)/(x)` in the given relation , we get
`g(x)g((1)/(x))=g(x)+g((1)/(x))=g(x)+g((1)/(x))+g((1)/(x))+g(1)-2`
`implies g(x)g((1)/(x))=g(x)+g((1)/(x))" "[ :' g(1)=2]`
`implies g(x)=x^(n)+1" " `
`implies g(2)=2^(n)+1 implies 5=2^(n)implies n=2 `
`:. g(x)=x^(n)+1 implies g(5)=25+1=26`.