If `lim_(xto oo) (1+(a)/(x)+(b)/(x^2))^(2x)=e^2`, then

Options

A. `a inR,bin R`
B. `a=1,b in R`
C. `a in R,b =2`
D. `a=1,b=2`

Answer

B

Explanation

`lim_(xtooo)(1+(a)/(x)+(b)/(x^2))^(2x)`
`=lim_(xtooo)(1+(a)/(x)+(b)/(x^2))^(2x((a//x+b//x^2)/(a//x+b//x^2)))`
`=e^((xtooo)^(lim2x(a//x+b//x^2))" "[because lim_(xtooo)(1+x)^(1//x)=e]`
`=e^(2^(lim2(a+b//x)))=e^(2a)` [given]
`rArr a=1 and b in R`