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Question
Evaluate the following limit :
`lim_(x -> 0) [((49)^x - 2(35)^x + (25)^x)/(sinx* log(1 + 2x))]`
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Solution
`lim_(x -> 0) [((49)^x - 2(35)^x + (25)^x)/(sinx* log(1 + 2x))]`
= `lim_(x -> 0) ((7^x)^2 - 2(7^x) (5^x) + (5^x)^2)/(sinx log (1 + 2x)) ...[(35^x = (5*7)^x = 5^x * 7^x),(49^x = (7^2)x = (7^x)^2)]`
= `lim_(x -> 0) [7^x - 5^x]^2/(sinx log (1 + 2x)`
= `lim_(x -> 0) [(7^x - 1) - (5^x- 1)]^2/(sin x log (1 + 2x))`
= `lim_(x -> 0) [(7^x - 1)/x - (5^x - 1)/x]^2/([sinx/x][(log(1 + 2x))/x])` ...[∵ x → 0, ∴ x ≠ 0]
= `([lim_(x -> 0) ((7^x - 1)/x - (5^x - 1)/x)]^2)/([lim_(x -> 0) sinx/x] xx 2 [lim_(x -> 0) (log(1 + 2x))/(2x)]`
= `[log 7 - log 5]^2/([1] xx 2[1]) ...[because x -> 0 "," 2x -> 0 "and" lim_(x -> 0) (log(1 + x))/x = 1 "and" lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
= `1/2[log(7/5)]^2`
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