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Question
Evaluate the following :
`lim_(x -> 0) [((5^x - 1)^2)/((2^x - 1)log(1 + x))]`
Solution
`lim_(x -> 0) ((5^x - 1)^2)/((2^x - 1)log(1 + x))`
= `lim_(x -> 0)((5^x - 1)^2/x^2)/(((2^x - 1)*log(1 + x))/x^2) ...[("Divide numerator and"),("denominator by" x^2.),(because x -> 0"," x ≠ 0),(therefore x^2 ≠ 0)]`
= `(lim_(x -> 0) ((5^x - 1)/x)^2)/(lim_(x -> 0) ((2^x - 1)/x)*(log(1 + x))/x`
= `(lim_(x -> 0) (5^x - 1)/x)^2/(lim_(x -> 0) ((2^x - 1)/x)*lim_(x -> 0) (log(1 + x))/x)`
= `(log5)^2/(log2 xx 1) ....[lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
= `(log5)^2/log2`
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