limitx>∞[(x^3+2x)^1/3]/(x^2+1)

[(x^3 + 2x)^(1/3)]/(x^2 + 1) = [x^3(1 + 2/x2)]^(1/3)]/[x^2(1 + 1/x^2)]
= [x(1 + 2/x2)^(1/3)]/[x^2(1 + 1/x^2)]
= (1 + 2/x^2)^(1/3)/[x(1 + 1/x^2)
When x tends to infinity, 2/x^2 and 1/x^2 tends to zero, the expression becomes 1/x. So limit tends to infinity is zero.