Question Number 104746 by bobhans last updated on 23/Jul/20

lim_(x→∞) (((1/2))^(3x) +((1/2))^x )^(1/x^2 )

Answered by Dwaipayan Shikari last updated on 23/Jul/20

lim_(x→∞) (((1/2))^(3x) +((1/2))^x )^(1/x^2 ) =y  (1/x^2 )log(((1/2))^(3x) +((1/2))^x )=logy  (1/x^2 )log(1+z−1).((z−1)/(z−1))=logy         {take z→0    as ((1/2))^x →0  ((((1/2))^(3x) +((1/2))^x −1)/x^2 )=logy  logy=0  y=1

Answered by mathmax by abdo last updated on 23/Jul/20

f(x) ={((1/2))^(3x)  +((1/2))^x }^(1/x^2 )   ⇒ln(f(x) =(1/x^2 )ln{((1/2))^(3x)  +((1/2))^x }  =(1/x^2 )ln{((1/2))^x (((1/2))^(2x)  +1)} =−(1/x)ln(2) +(1/x^2 )ln(1+(1/4^x )) ⇒  ln(f(x)) ∼−((ln(2))/x) +(1/(x^2 .4^x )) (x→+∞) ⇒lim_(x→+∞) ln(f(x)) =0 ⇒lim_(x→+∞) f(x)=1