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Question
Select the correct answer from the given alternatives:
If f(x) = `(12^x - 4^x - 3^x + 1)/(1 - cos 2x)`, for x ≠ 0 is continuous at x = 0 then the value of f(0) is ______.
Options
`log12/2`
log2.log3
`(log2*log3)/2`
None of these.
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Solution
If f(x) = `(12^x - 4^x - 3^x + 1)/(1 - cos 2x)`, for x ≠ 0 is continuous at x = 0 then the value of f(0) is log 2 · log 3.
Explanation:
If f(x) is continuous at x = 0 (given)
`therefore "f"(0) = lim_(x -> 0) "f"(x)`
`= lim_(x -> 0) (12^x - 4^x - 3^x + 1)/(1 - cos 2x)`
`= 1/2 lim_(x ->0) (4^x (3^x - 1)(3^x - 1))/(sin^2 x)`
`= 1/2 lim_(x -> 0) ((3^x - 1)(4^x - 1))/(sin^2 x)`
`= 1/2 (lim_(x -> 0) ((3^x - 1)/x) * lim_(x - > 0) ((4^x - 1)/x))/(lim_(x -> 0) (sin x)/x)^2`
`= 1/2 xx ((log 3) xx (log 4))/(1)^2`
`= 1/2 xx log 3 xx log (2)^2`
= log 3 · log 2
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