limn 3rd root of n3an2bn3 root of n2n1SolutionLt ninfinity

lim(n->?) [3rd root of (n^3+an^2+bn+3) + root of (n^2+n+1)]

Solution

Lt n-->infinity {n^3+an^2+bn+3)^(1/3)/ (n^2+n+1)^(1/2)

=Lt n-->inf {(n^3+an^2+bn+3)^(1/3)}/{n^2+n+1)^(1/2) = lt {(n^3+an^2+bn+3)^(2/6)}/{n^2+n+1)^(3/6)}

=lt n--> inf{ [(n^3+an^2+bn+3)^2]/[n^2+n+1)^3]}^(1/6)= lt {[1 +a/n+b/n^2+3/n^3]}^2 /[1+1/n+1/n^2)^3}^(1/6)

= {(1+a*0+3*0)^2 / (1+0+0)^3}^(1/6)

= 1/1

=1

Therefore lt n--> inf{ [(n^3+an^2+bn+3)^2]/[n^2+n+1)^3]}^(1/6)= 1.