if `lim_(xto0) (ae^(2x)-bcos2x+ce^(-2x)-xsinx)/(xsinx)=1` and `f(t)=(a+b)t^(2)+(a-b)t+c`, then

Options

A. `a=(1)/(4)`
B. `c=(1)/(2)`
C. `f(1)=(3)/(4)`
D. `f(1)=1`

Answer

A, C

Explanation

`underset(xto0)lim(ae^(2x)-bcos2x+ce^(-2x)-xsinx)/(xsinx)=1`
`underset(xto0)lim((1+2x+((2x)^(2))/(2!)+....)-b(1+((2x)^(2))/(2!)+((2x)^(4))/(4!)-....)+c(1-2x+((2x)^(2))/(2!)-....))/(x^(2))`
`=2`
`impliesa-b+c=0` ..(1)
`2a-2c=0` . . .(2)
`2a+2b+2c=2` ..(3)
`impliesa=c` and `b=2a`
`impliesa+2a+a=1impliesa=1//4=c`
`b=(1)/(2)`
`f(t)=(3)/(4)t^(2)-(1)/(4)t+(1)/(4)=(3t^(2)-t+1)/(4)`