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If f(x) = 24x-8x-3x+112x-4x-3x+1 for x ≠ 0 = k, for x = 0 is continuous at x = 0, find k. - Mathematics and Statistics

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Sum

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

Function f 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))`

= `lim_(x→0) (8^x - 1)/(4^x - 1)  [(because x→0","   3^x → 3^0),(therefore 3^x → 1 therefore 3^x ≠ 1),(therefore 3^x - 1 ≠ 0)]`

= `lim_(x→0) (((8^x - 1)/x)/((4^x - 1)/x))`  .....[∵ x → 0, ∴ x ≠ 0]

= `log 8/log 4   ...[because  lim_(x→0) (("a"^x - 1)/x) = log"a"]`

= `log(2)^3/log(2)^2`

= `(3log2)/(3log2)`

∴ f(0) = `3/2`

Concept: Continuity in the Domain of the Function
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