If `lim_(xrarroo) xlog_(e)(|(alpha//x,1,gamma),(0,1//x,beta),(1,0,1//x)|)=-5.` then where `alpha,beta,gamma` are finite real numbers, then a) `alpha=2,beta=1,gamma` in `R` b) `alpha=2,beta=2,gamma=5` c) `alpha` in `R` ,`beta=1`,`gamma` in `R` d) `alpha` in `R` ,`beta=1,gamma=5`

Answer

A, B, C

Explanation

`underset(xrarr0)(lim)(ae^(x)+bcosx+ce^(-x))/(e^(2x)-2e^(x)+1)`
`rArr underset(xrarr0)(lim)(a(1+x+(x^(2))/(2)+...)+b(1-(x^(2))/(2)...)+c(1-x+(x^(2))/(2)))/(x^(2)).((x)/(e^(x)-1))^(2)=4`
`rArr" "a+b+c=0, a-c=0 rArr a=c`
`rArr" "(a-b+c)/(c)=4`
`rArr" "a-b+c=8 rArr 2b = -8 rArr b=-4`