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Question
Evaluate the following limit :
`lim_(x -> 0)[(15^x - 5^x - 3^x + 1)/(x*sinx)]`
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Solution
`lim_(x -> 0)[(15^x - 5^x - 3^x + 1)/(xsinx)]`
= `lim_(x -> 0) (5^x * 3^x - 5^x - 3^x + 1)/(xsinx)`
= `lim_(x -> 0) (5^x (3^x - 1) - (3^x - 1))/(xsinx)`
= `lim_(x -> 0) ((3^x - 1)(5^x - 1))/(xsinx)`
= `lim_(x -> 0) (((3^x - 1)/x)((5^x - 1)/x))/((sinx/x))` ...[∵ x → 0, ∴ x ≠ 0]
= `((log3)(log5))/1 ...[because lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
= (log 3) (log 5).
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