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If lim_(xrarroo) xlog_(e)(|(alpha//x,1,gamma),(0,1//x,beta),(1,0,1//x)|)=-5. where alpha, beta, gamma are finite real numbers, then

Answer»

`alpha=2, BETA=1, gamma in R`
`alpha=2, beta=2, gamma=5`
`alpha in R, beta =1, gamma in R`
`alpha in R, beta = 1, gamma=5`

Solution :Let, `L=UNDERSET(xrarroo)(lim)X.log_(e)(|(alpha//x,1,gamma),(0,1//x,beta),(1,0,1//x)|)`
`=underset(xrarroo)(lim)x.log_(e)((alpha)/(x^(3))+beta-(gamma)/(x))`
For LIMIT to EXIST, `beta=1`
`therefore""L=underset(xrarroo)(lim)x.log_(e)(2+(alpha)/(x^(3))-(gamma)/(x))`
`=underset(xrarroo)(lim)x((alpha)/(x^(3))-(gamma)/(x))`
`=underset(xrarroo)(lim)((alpha)/(x^(2))-gamma)=-gamma=-5 and alpha in R`


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