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Question
Evaluate the following limit :
`lim_(x -> 0) [(9^x - 5^x)/(4^x - 1)]`
Solution
`lim_(x -> 0) (9^x - 5^x)/(4^x - 1) = lim_(x -> 0) (9^x - 1 + 1 - 5^x)/(4^x - 1)`
= `lim_(x -> 0) ((9^x - 1) - (5^x - 1))/(4^x - 1)`
= `lim_(x -> 0) ((9^x - 1 - 5^x - 1)/x)/((4^x - 1)/x` ...[∵ x → 0, ∴ x ≠ 0]
= `lim_(x -> 0) (((9^x - 1)/x) - ((5^x - 1)/x))/(((4^x - 1)/x))`
= `(lim_(x -> 0) (9^x - 1)/x - lim_(x -> 0) (5^x - 1)/x)/(lim_(x -> 0) (4^x - 1)/x`
= `(log9 - log5)/log4 ...[because lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
= `1/((log4)) log(9/5)`
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