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Question
Evaluate the following limit :
`lim_(x -> 0) [(6^x + 5^x + 4^x - 3^(x + 1))/sinx]`
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Solution
`lim_(x -> 0) [(6^x + 5^x + 4^x - 3^(x + 1))/sinx]`
= `lim_(x -> 0) [((6^x - 1) + (5^x - 1) + (4^x - 1) - 3^(x + 1) + 3)/sinx]`
= `lim_(x -> 0) ((6^x - 1) + (5^x - 1) + (4^x - 1) - 3(3^x - 1))/sinx`
= `lim_(x -> 0) (((6^x - 1)/x) + ((5^x - 1)/x) + ((4^x - 1)/x) - 3((3^x - 1)/x))/((sinx/x)` ...[∵ x → 0 ∴ x ≠ 0]
= `(lim_(x -> 0) (6^x - 1)/x + lim_(x -> 0) (5^x - 1)/x + lim_(x -> 0) (4^x - 1)/x - 3 lim_(x -> 0)(3^x - 1)/x)/((lim_(x -> 0) sinx/x))`
= `(log 6 + log 5 + log 4 - 3 log 3)/1 ...[because lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
= log(6 × 5 × 4) – log 33
= `log((6 xx 5 xx 4)/27)`
= `log(40/9)`.
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