The tutor uses l’Hôpital’s rule to find a limit of form ∞/∞.

l’Hôpital’s rule states that the limit of a quotient of form ∞/∞ or 0/0 can be found as follows:

lim (f(x)/g(x)) = lim (f'(x)/g'(x))

In this case [noting the square root of x is x^0.5]:

lim_x→∞(lnx/x^0.5) = (by l’Hôpital) lim_x→∞((x^-1)/(0.5x^-0.5))

which becomes

lim_x→∞2x^-0.5 or lim_x→∞2/x^0.5 = 0

By that reasoning, the reciprocal limit, lim_x→∞(x^0.5/lnx), should not exist.

Source:

Larson, Roland E. and Robert P. Hostetler. Calculus, 3rd ed. Toronto: DC Heath, 1989.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.