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Question
Evaluate the following limit :
`lim_(x -> 0) [(5^x + 3^x - 2^x - 1)/x]`
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Solution
`lim_(x -> 0) [(5^x + 3^x - 2^x - 1)/x]`
= `lim_(x -> 0) [((5^x - 1) + (3^x - 1) - (2^x - 1))/x]`
= `lim_(x -> 0) [(5^x - 1)/x + (3^x - 1)/x - (2^x - 1)/x]`
= `lim_(x -> 0) (5^x - 1)/x + lim_(x -> 0) (3^x - 1)/x - lim_(x -> 0) (2^x - 1)/x`
= log 5 + log 3 – log 2 ...`[because lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
= `log((5 xx 3)/2)`
= `log(15/2)`
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