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

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Answer 1 · 2016-11-10T17:55:03

$e^6$

$((x+1)/(x-2))=((x-2)+3)/(x-2)=1+3/(x-2)$ now calling $y = (x-2)/3$ we have

$((x+1)/(x-2))^(2x-1)=(1+1/y)^(6y+3)$ so

$lim_(x->oo)((x+1)/(x-2))^(2x-1)=lim_(y->oo)(1+1/y)^(6y)lim_(y->oo)(1+1/y)^3 = (lim_(y->oo)(1+1/y)^y)^6 (1+lim_(y->oo)1/y)^3=e^6$

lim_(x->oo)((x+1)/(x-2))^(2x-1)=lim_(x->oo)((x+1)/(x-2))^(2x-1)= ?