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Question
Select the correct answer from the given alternatives.
`lim_(x -> 0) [(log(5 + x) - log(5 - x))/sinx]` =
Options
`3/2`
`-5/2`
`-1/2`
`2/5`
Solution
`2/5`
Explanation;
`lim_(x -> 0) [(log(5 + x) - log(5 - x))/sinx]`
= `lim_(x -> 0)(log[5(1 + x/5)] - log[5(1 - x/5)])/sinx`
= `lim_(x -> 0) (log5 + log(1 + x/5) - [log5 + log(1 - x/5)])/sinx`
= `lim_(x -> 0)[(log(1 + x/5) -log(1 - x/5))/x xx x/sinx]`
= `lim_(x -> 0) [log(1 + x/5)/(5(x/5)) - (log(1 - x/5))/((-5)((-x)/5))] xx lim_(x -> 0) x/sinx`
= `[1/5 (1) + 1/5(1)] xx 1`
= `2/5`
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