Question #27259

2 Answers
Dec 20, 2017

e^(-4)e4

Explanation:

Lim_(xrarroo)((3x-4)/(3x+2))^(2x-4)

Lim_(xrarroo)((3x+2-6)/(3x+2))^(2x-4)

Lim_(xrarroo)((3x+2)/(3x+2)+(-6)/(3x+2))^(2x-4)

Lim_(xrarroo)(1+1/((3x+2)/(-6)))^(2x-4)

Lim_(xrarroo)(1+1/((3x+2)/(-6)))^((2x-4)(3x+2)/(-6)*(-6)/(3x+2))

Lim_(xrarroo)[(1+1/((3x+2)/(-6)))^((3x+2)/(-6))]^((2x-4)*(-6)/(3x+2))


Lim_(xrarroo)(1+1/x)^x=e


e^(Lim_(xrarroo)((-12x+24)/(3x+2)))

e^(-12/3)=e^(-4)

Dec 20, 2017

e^-4

Explanation:

Making

1+1/y = ((3 x - 4)/(3 x + 2)) or equivalently

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

((3 x - 4)/(3 x + 2))^(2 x - 4) equiv (1+1/y)^(-(4y+16/3))=1/( (1+1/y)^(4y+16/3))

and then

lim_(x->oo)((3 x - 4)/(3 x + 2))^(2 x - 4) =1/( lim_(y->oo)(1+1/y)^(4y+16/3)) = 1/e^4 = e^-4