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Evaluate: lim(x->0)(log(5+x)-"log"(5-x))...

Evaluate: `lim_(x->0)(log(5+x)-"log"(5-x))/x`

Text Solution

Verified by Experts

We have `underset(xto0)lim(log(5+x)-log(5-x))/(x)" "`(0/0 form)
`=underset(xto0)lim(log{5(1+(x)/(5))}-log{5(1-(x)/(5))})/(x)`
`=underset(xto0)lim({log5+log(1+(x)/(5))}-{log5+log(1- (x)/(5))})/(x)`
`=underset(xto0)lim(log(1+(x)/(5))-log(1-(x)/(5)))/(x)`
`=underset(xto0)lim(1)/(5)(log(1+(x)/(5))^(x))/(x//5)+underset(xto0)limlog(1-(x)/(5))/(-x//5)1/5=1/5+1/5=2/5`
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