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Question
Find a and b if following function is continuous at the point or on the interval indicated against them:
f(x) `{:(= (4tanx + 5sinx)/("a"^x - 1)",", "for" x < 0),(= (9)/(log2)",", "for" x = 0),(= (11x + 7x*cosx)/("b"^x - 1)",", "for" x > 0):}`
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Solution
f(x) is continuous at x = 0
∴ `lim_(x -> 0^-) "f"(x)` = f(0)
∴ `lim_(x -> 0) [(4tanx + 5sinx)/("a"^x - 1)] = 9/log2`
∴ `lim_(x -> 0) [((4tanx + 5sinx)/x)/(("a"^x - 1)/x)]` ...[∵ x → 0, x ≠ 0]
= `9/log2`
∴ `(lim_(x -> 0)((4tanx)/x + (5sinx)/x))/(lim_(x -> 0) ("a"^x - 1)/x) = 9/log2`
∴ `(4lim_(x -> 0) (tanx)/x + 5 lim_(x -> 0) (sinx)/x)/(lim_(x -> 0) ("a"^x - 1)/x) = 9/log2`
∴ `(4(1) + 5(1))/(log"a") = 9/log2 ...[because lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
∴ `9/log"a" = 9/log2`
∴ log a = log 2
∴ a = 2
Also `lim_(x -> 0^+) "f"(x)` = f(0)
∴ `lim_(x -> 0) (11x + 7x*cosx)/("b"^x - 1) = 9/log2`
∴ `lim_(x -> 0) ((11x + 7x cosx)/x)/(("b"^x - 1)/x) = 9/log2` ...[∵ x → 0, x ≠ 0]
∴ `(lim_(x -> 0)(11 + 7cosx))/(lim_(x -> 0)(("b"^x - 1)/x)) = 9/log2`
∴ `(11 + 7cos0)/log"b" = 9/log2 ...[because lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
∴ `(11 + 7(1))/log"b" = 9/log2`
∴ 9log b = 18log 2
∴ log b = 2log 2
= log(2)2
∴ log b = log 4
∴ b = 4
∴ a = 2 and b = 4
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