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Question
Evaluate the following Limits: `lim_(x -> 0) [("a"^(4x) - 1)/("b"^(2x) - 1)]`
Solution
`lim_(x -> 0) ("a"^(4x) - 1)/("b"^(2x) - 1)`
= `lim_(x -> 0) (("a"^(4x) - 1)/x)/(("b"^(2x) - 1)/x)`
= `(lim_(x -> 0)(("a"^(4x) - 1)/(4x)) xx 4)/(lim_(x -> 0)(("b"^(2x) - 1)/(2x)) xx 2`
= `(4log"a")/(2log"b") ...[("As" x -> 0"," 2x -> 0"," 4x -> 0),(and lim_(x -> 0) ("a"^x - 1)/x = log "a")]`
= `(2log"a")/(log "b")`
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