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Question
Evaluate the following: `lim_(x -> 0)[((49)^x- 2(35)^x + (25)^x)/x^2]`
Solution
`lim_(x -> 0)[((49)^x- 2(35)^x + (25)^x)/x^2]`
= `lim_(x-> 0) [((7^2)^x - 2(7 xx 5)^x + (5^2)^x)/x^2]`
= `lim_(x-> 0) [((7^x)^2 - 2 * 7^x xx 5^x + (5^x)^2)/x^2]`
= `lim_(x -> 0) [(7^x - 5^x)^2/x^2]`
= `lim_(x -> 0) ((7^x - 5^x)/x)^2`
= `lim_(x -> 0) [(7^x - 1 - 5^x - 1)/x]^2`
= `lim_(x -> 0)[(7^x - 1)/x - (5^x - 1)/x]^2`
= `[lim_(x -> 0) (7^x - 1)/x - lim_(x -> 0) (5^x - 1)/x]^2`
= `(log 7 - log 5)^2 ...[lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
= `(log 7/5)^2`.
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