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Question
Select the correct answer from the given alternatives:
f(x) `{:(= (32^x - 8^x - 4^x + 1)/(4^x - 2^(x + 1) + 1)",", "for" x ≠ 0),(= "k""," , "for" x = 0):}` is continuous at x = 0, then value of ‘k’ is
Options
6
4
(log 2)(log 4)
3 log 4
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Solution
6
Explanation;
f(x) is continuous at x = 0
∴ f(0) = `lim_(x -> 0) "f"(x)`
∴ k = `lim_(x -> 0) (32^x - 8^x - 4^x + 1)/(4^x - 2^(x + 1) + 1)`
= `lim_(x -> 0) ((4^x - 1)(8^x - 1))/(2^x - 1)^2`
= `(lim_(x -> 0)((4^x - 1)/x)((8^x - 1)/x))/(lim_(x -> 0)((2^x - 1)/x)^2`
= `(lim_(x -> 0)((4^x - 1)/x) * lim_(x -> 0)((8^x - 1)/x))/((lim_(x -> 0) (2^x - 1)/x)^2`
= `(log4 xx log8)/(log2)^2`
= `(2log2 xx 3log2)/(log2)^2`
= 6
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