How to calculate this? $lim_(x->oo)(ln((x+2)/x)-ln((x+1)/(x-1))+1)$

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Answer 1 · 2017-04-25T19:32:50

$1$

$lim_(x->oo)(ln((x+2)/x) - ln((x+1)/(x-1)) + 1)$

$=lim_(x->oo)(ln(1+2/x) - ln(1+2/(x-1)) + 1)$

$=ln(1+0) - ln(1+0) + 1$

$=0-0+1$

$=1$

Answer 2 · 2017-04-25T19:35:32

$1$

$ln((x+2)/x)-ln((x+1)/(x-1))+1 = ln((e (x + 2) (x - 1))/(x(x+1)))$

and

$lim_(x->oo)(ln((x+2)/x)-ln((x+1)/(x-1))+1) = lim_(x->oo)ln((e (x + 2) (x - 1))/(x(x+1)))=$
$= ln(lim_(x->oo)(e (x + 2) (x - 1))/(x(x+1))) =$

$= ln(lim_(x->oo)(e (1 + 2/x) (1 - 1/x))/((1+1/x))) = ln(e xx 1)=1$

How to calculate this?lim_(x->oo)(ln((x+2)/x)-ln((x+1)/(x-1))+1)lim_(x->oo)(ln((x+2)/x)-ln((x+1)/(x-1))+1)