What is
lim (t^3 + 1)/(t lnt)
x->infinity
lim (t^3+1)/(t lnt) as t>inf
first multiply numerator and denominator by 1/t
(t^2+1/t)/lnt you can see that the 1/tlnt term is vanishingly small.
So you are left with t^2/lnt
Now take the anti log of numerator, denominator, it will have the same limit
e^(t^2)/t which clearly has a limit of inf.
You can also use L'Hopitals rule
lim indetminate/indeterminate=
lim d(numerator)/d(denominator)
so
lim (t^3+1)/tlnt= 3t^2/(lnt+1)=6t/(1/t)=6t^2 = inf