Question

How do you show that `lim_(x->\infty)(2^(x+3)-2*3^(x-1))/(2^(x-1)+3^(x-2))=-6`?

How do you show that `lim_(x->\infty)(2^(x+3)-2*3^(x-1))/(2^(x-1)+3^(x-2))=-6`?

Answer 1

See below.

Explanation:

`(2^(x+3)-2*3^(x-1))/(2^(x-1)+3^(x-2)) = 3^x/3^x((2^3(2/3)^x-2/3)/(1/2(2/3)^x+1/9)) = (2^3(2/3)^x-2/3)/(1/2(2/3)^x+1/9)`

Now

`lim_(x->oo)(2^(x+3)-2*3^(x-1))/(2^(x-1)+3^(x-2)) = (lim_(x->oo)2^3(2/3)^x-2/3)/(1/2lim_(x->oo)(2/3)^x+1/9) = (-2/3)/(1/9) = -6`