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Question
Select the correct answer from the given alternatives.
`lim_(x→0)[(3^(sinx) - 1)^3/((3^x - 1).tan x.log(1 + x))]` =
Options
3log 3
2log 3
(log 3)2
(log 3)3
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Solution
`lim_(x→0)[(3^(sinx) − 1)^3/((3^x − 1).tan x.log(1 + x))]` = (log 3)2
Explanation:
`lim_(x→0)[(3^(sinx) - 1)^3/((3^x - 1).tan x.log(1 + x))]`
`= (lim_(x →0) (3^(sin x) - 1)^3/(sin^3 x). (sin^3 x)/(x^3))/(lim_(x →0) ((3^x − 1)/x).(tan x/x).log(1 + x)/x)`
`= (lim_(x →0) ((3^(sinx) - 1)/sin x)^3. lim_(x →0) (sin x/x)^3)/(lim_(x →0) ((3^x − 1)/x). lim_(x →0) (tan x/x). lim_(x →0) (log(1 + x)/x))`
`= ((log 3)^3 × (1)^3)/(log 3 × 1 × 1)`
`= (log 3)^3/(log 3)`
= (log 3)2
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