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Question
Evaluate the following limits:
`lim_(x -> 0) (3^x - 1)/(sqrt(x + 1) - 1)`
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Solution
We know `lim_(x -> 0) ("a"^x - 1)/x = log "a", "a" > 0`
`lim_(x -> 0) ((3^x- 1)/(sqrt(x + 1) - 1)) = lim_(x -> 0) ((3^x - 1)/(sqrt(x + 1) - 1)) xx (sqrt(x + 1) + 1)/(sqrt(x + 1) + 1)`
= `lim_(x -> 0) ((3^x - 1) (sqrt(x + 1) + 1))/(x + 1 - 1)`
= `lim_(x -> 0) ((3^x - 1) (sqrt(x + 1) + 1))/x`
= `lim_(x -> 0) ((3^x - 1)/x) xx lim_(x -> 0) (sqrt(x + 1) + 1)`
= `log 3 xx (sqrt(0 + 1) + 1)`
= `log 3 xx (1 + 1)`
= 2 log 3
= log 32
`lim_(x -> 0) (3^x - 1)/(sqrt(x + 1) - 1) = log 9`
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