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Question
If f(x) `{:(= (24^x - 8^x - 3^x + 1)/(12^x - 4^x - 3^x + 1)",", "for" x ≠ 0), (= "k"",", "for" x = 0):}}` is continuous at x = 0, find k
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Solution
f(x) is continuous at x = 0
∴ f(0) = `lim_(x -> 0) "f"(x)`
∴ k = `lim_(x -> 0) (24^x - 8^x - 3^x + 1)/(12^x - 4^x - 3^x + 1)`
= `lim_(x -> 0) (8^x * 3^x - 8^x - 3^x + 1)/(4^x * 3^x - 4^x - 3^x + 1)`
= `lim_(x -> 0) (8^x(3^x - 1) - 1(3^x - 1))/(4^x (3^x - 1) - 1(3^x - 1)`
= `lim_(x -> 0) ((3^x - 1)(8^x - 1))/((3^x - 1)(4^x - 1)) ...[(because x -> 0"," 3x -> 3^0),(therefore 3^x -> 1 therefore 3^x ≠ 1),(therefore 3^x - 1 ≠ 0)]`
= `lim_(x -> 0) (8^x - 1)/(4^x - 1)`
= `lim_(x -> 0) (((8^x - 1)/x)/((4^x - 1)/x))` ...[∵ x → 0, ∴ x ≠ 0]
= `(lim_(x -> 0) (8^x - 1)/x)/(lim_(x -> 0) (4^x - 1)/x)`
= `log8/log4 ...[because lim_(x -> 0) (("a"^x - 1)/x) = log"a"]`
= `log(2)^3/log(2)^2`
= `(3log2)/(2log2)`
∴ f(0) = `3/2`
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