Question #27259

2 Answers
Dec 20, 2017

#e^(-4)#

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#