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Question
Evaluate the following limit :
`lim_(x ->0) [("a"^x - "b"^x)/(sin(4x) - sin(2x))]`
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Solution
`lim_(x ->0) [("a"^x - "b"^x)/(sin(4x) - sin(2x))]`
=`lim_(x - 0)[(("a"^x - 1) - ("b"^x - 1))/(sin4x - sin2x)]`
= `lim_(x -> 0) (("a"^x - 1)/x - ("b"^x - 1)/x)/(4* (sin4x)/(4x) - 2* (sin2x)/(2x))` ...[∵ x→0, ∴ x ≠ 0]
= `(lim_(x -> 0) ("a"^x - 1)/x - lim_(x -> 0) ("b"^x - 1)/x)/(4 lim_(x -> 0) (sin4x)/(4x) - 2 lim_(x -> 0) (sin2x)/(2x))`
= `(log"a" - log"b")/(4(1) - 2(1)) ...[because x -> 0, 4x -> 0, 2x -> 0 "and" lim_(theta -> 0) sintheta/theta = 1]`
= `1/2 log ("a"/"b")`
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