If `f(x)` be a twice differentiable function from `RR rarr RR` such that `t^(2)f(x)-2tf'(x)+f''(x)=0` has two equal values of for all `x` and `f(0)=1,f'(0)=2,` then `lim_(x rarr 0)((f(x)-1)/(x)-(t)/(2))` is

Answer

1

Explanation

`t^(2)f(x)-2tf^(')(x)+f^('')(x)=0` has equal roots
`D=0implies(f^('')(x))/(f^(')(x))=(f^(')(x))/(f(x))`
`impliesIn (f^(')(x))=In (f(x))-InC`
`impliesf(x)=Cf^(')(x)`
`impliesf(0)=C.f^(')(0)`
`impliesC=1/2`
`:.(f^(')(x))/(f(x))=2impliesInf(x)=2x+k`
`impliesIn (f(0))=kimpliesk=0`
`impliesIn (f(x))=2ximpliesf(x)=e^(2x)`
`t^(2)e^(2x)-4te^(2x)+4e^(2x)=0`
`lim_(x to 0) ((f(x)-1)/x-t/2)=lim_(x to o) ((e^(2x)-1)/(2x)xx 2- 2/2)=1`