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Question
Evaluate the following Limits: `lim_(x -> 0)[(log(4 - x) - log(4 + x))/x]`
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Solution
`lim_(x -> 0)(log(4 - x) - log(4 + x))/x`
= `lim_(x -> 0) (log[4(1 - x/4)] - log[4(1 + x/4)])/x`
= `lim_(x -> 0)(log4 + log(1 - x/4) - [log4 log(1 + x/4)])/x`
= `lim_(x -> 0) (log(1 - x/4) - log(1 + x/4))/x`
= `lim_(x -> 0)[(log(1 - x/4))/x - (log(1 + x/4))/x]`
= `lim_(x -> 0) (log(1 - x/4))/((-4)(-x/4)) - lim_(x -> 0) (log(1 + x/4))/(4(x/4)`
= `-1/4 lim_(x -> 0) (log(1 - x/4))/(-x/4) - 1/4 lim_(x -> 0) (log(1 + x/4))/(x/4)`
= `-1/4(1) - 1/4(1) ...[("As" x -> 0"," x/4 -> 0"," (-x)/4 _> 0),(and lim_(x -> 0) (log(1 + x))/x = 1)]`
= `-1/2`
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