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