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Let a gt 0 be real number. Then the limi...

Let a `gt` 0 be real number. Then the limit
`lim_(xto2) (a^(x)+a^(3-x)-(a^(2)+a))/(a^(3-x)-a^(x//2))`
is-

A

2 log a

B

`-(4)/(3)` a

C

`(a^(2)+a)/(2)`

D

`(2)/(3)(1-a)`

Text Solution

Verified by Experts

The correct Answer is:
D
`underset(xto2)lim(a^(x)+a^(3-x)-(a^(2)+a))/(a^(3-x)-a^(x//2))((0)/(0))`
Apply L hospital rule
`underset(xto2)lim(a^(x)lna-a^(3-x)lna)/(-a^(3-x)lna-(1)/(2)a^(x//2lna))`
`(a^(2)lna-alna)/(-alna-(a)/(2)lna)=(a^(2)-a)/(-(3a)/(2))=(2)/(3)(1-a)`
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