1.

Consider f(x)=sin^(-1)((x+3)/(2x+5)),g(x)=sin^(-1)((ax^(2)+b)/(x^(2)+5)). If Lim_(xto oo)(f(x)-g(x))=0 and Lim_(xto oo) (f(x)+g(x))=(pi)/(4), then find the value of (a+b^(2)).

Answer»

Solution :`underset(xtooo)Limsin^(-1)((x+3)/(2x+5))-SIN^(-1)((ax^(2)+b)/(x^(2)+5))=0"as"xtooosin^(-1)((x+3)/(2x+5))=(PI)/(6)`
`:.""underset(xtooo)Limsin^(-1)((ax^(2)+b)/(x^(2)+5))=(pi)/(6)"":.""a=(1)/(2)`
Now,
`underset(xtooo)Limsin^(-1)((x+3)/(2x+5))+sin^(-1)((ax^(2)+b)/(x^(2)+5))=(pi)/(4)rArr"sin"^(-1)(3)/(4)rArr+"sin"^(-1)((b)/(5))=(pi)/(4)`
`sin^(-1)((b)/(5))=tan^(-1)1-"tan"^(-1)(3)/(4)rArr""sin^(-1)((b)/(5))=sin^(-1)((1)/(SQRT(50)))`
`:.""b=(1)/(sqrt(2))"":.""b^(2)=(1)/(2)`
`:.""a+b^(2)=1`.


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