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Question
Evaluate the following: `lim_(x -> 0)[(log(2 + x) - log( 2 - x))/x]`
Solution
`lim_(x -> 0)[(log(2 + x) - log( 2 - x))/x]`
= `lim_(x -> 0)(log[2(1 + x/2)] - log[2(1 - x/2)])/x`
= `lim_(x -> 0) (log2 + log(1 + x/2) - [log2 + log(1 - x/2)])/x`
= `lim_(x -> 0) (log(1 + x/2) - log(1 - x/2))/x`
= `lim_(x -> 0) [(log(1 + x/2))/x - (log(1 - x/2))/x]`
= `lim_(x -> 0) [(log(1 + x/2))/(2(x/2)) - (log(1 - x/2))/((-2)(-x/2))]`
= `1/2 lim_(x -> 0) (log(1 + x/2))/(x/2) + 1/2 lim_(x -> 0) (log(1 - x/2))/(-x/2)`
= `1/2(1) + 1/2(1) ....[(because x -> 0"," x/2 -> 0 and),(lim_(x -> 0) (log(1 + x))/x = 1)]`
= 1
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