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Question
Evaluate the following limit :
`lim_(x -> 0)[(2^x - 1)^3/((3^x - 1)*sinx*log(1 + x))]`
Solution
`lim_(x -> 0)(2^x - 1)^3/((3^x - 1)*sinx*log(1 + x))`
= `lim_(x -> 0)((2^x - 1)^3/x^3)/(((3^x - 1)*sinx*log(1 + x))/x^3) ...[("Divide numerator and"),("denominator by" x^3.),(because x -> 0"," x ≠ 0),(therefore x^3 ≠ 0)]`
= `(lim_(x -> 0) ((2^x - 1)/x)^3)/(lim_(x -> 0) ((3^x - 1)/x)* sinx/x* (log(1 + x))/x`
= `(lim_(x -> 0) (2^x - 1)/x)^3/((lim_(x -> 0) (3^x - 1)/x)*(lim_(x -> 0)sinx/x)(lim_(x -> 0) (log(1 + x))/x)`
= `(log2)^3/((log3)(1)(1)) ....[lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
= `(log2)^3/log3`
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