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