Question

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

Answer 1

`e^6`

Explanation:

`((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`