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Question
Evaluate the following: `lim_(x -> 0)[(15^x - 5^x - 3^x +1)/x^2]`
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Solution
`lim_(x -> 0)[(15^x - 5^x - 3^x +1)/x^2]`
= `lim_(x -> 0) (5^x*3^x - 5^x - 3^x + 1)/x^2`
= `lim_(x -> 0) (5^x (3^x- 1) - 1(3^x - 1))/x^2`
= `lim_(x -> 0) ((3^x - 1) (5^x - 1))/x^2`
= `lim_(x -> 0) ((3^x - 1)/x xx (5^x - 1)/x)`
= `lim_(x -> 0) (3^x - 1)/x xx lim_(x -> 0)(5^x - 1)/x`
= `log 3* log 5 ...[lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
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