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Answer 1 · 2017-03-31T19:21:00

#0.#

The Reqd. Lim#=lim_(x to oo)[(2x+2)/(3x-1)]^(3x)#

#=lim {((2x)(1+1/x))/((3x)(1-1/(3x)))]^(3x)#

#=[lim {(2/3)^3}^x][lim {(1+1/x)^x}^3]/[lim {(1+(1/(-3x)))^(-3x)}^(-1)#

#=[lim(8/27)^x][lim {(1+1/x)^x}^3]/[lim {(1+(1/(-3x)))^(-3x)}^(-1)#

Now, recall the following Standard Limits :

#(1) : lim_(y to oo) r^y=0, if |r|<1.#

#(2) : lim_(y to oo) (1+1/y)^y=e.#

#:." The Reqd. Lim="0*[e^3]/[e^(-1)]=0*e^4=0.#

Enjoy Maths.!