AP EAMCET202125 Aug 2021Evening ShiftMathematicsLimitsActual
If _ x 0 1+x (1+a^2 ) ^ 1 / x =2 a ^2 , a>0 and R , then
Options
- A=n 2 ,(n Z)
- B=2 n 2 ,(n Z)
- C=n + 2 ,(n Z)
- D=n 4 ,(n Z)
Correct answer
A. =n 2 ,(n Z)
Step-by-step solution
_ x 0 1+x (1+a^2 ) ^ 1 / x =2 a ^2 , a>0 and R LHS _ x 0 1+x (1+a^2 ) ^ 1 / x is of the form 1^ aligned & e^ _ x 0 1 x 1+x (1+a^2 )-1 e^ _ x 0 (1+a^2 ) & 1+a^2=2 a ^2 a^2-2 a ^2 +1=0 aligned aligned a & = 2 ^2 4 ^4 -4 2 a & = 2 ^2 4 ( ^4 -1 ) 2 a & = ^2 ^4 -1 aligned aligned & a>0 & ^4 =1 & ^2 =1= ^2 2 & =n 2 aligned