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If lim(xrarroo) xlog(e)(|(alpha//x,1,gam...

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

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`

Text Solution

Verified by Experts

The correct Answer is:
D
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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