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Question
Evaluate the following Limits: `lim_(x -> 0)[(x(6^x - 3^x))/((2^x - 1)*log(1 + x))]`
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Solution
`lim_(x -> 0)(x(6^x - 3^x))/((2^x - 1)*log(1 + x))`
= `lim_(x -> 0)(x(3^x*2^x - 3^x))/((2^x - 1)*log(1 + x))`
= `lim_(x -> 0) (x*3^x(2^x - 1))/((2^x - 1)*log(1 + x)`
= `lim_(x -> 0) (x*3^x)/(log (1 + x)) ...[("As" x -> 0"," 2^x -> 2^0),("i.e." 2^x -> 1 therefore 2^x ≠ 1),(therefore 2^x - 1 ≠ 0)]`
= `lim_(x -> 0) (3^x)/((log(1 + x))/x`
= `(lim_(x -> 0) 3^x)/(lim_(x -> 0) (log(1 + x))/x`
= `3^0/1 ...[lim_(x -> 0) (log(1 + x))/x = 1]`
= 1
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