Calculus: l’Hôpital’s rule: lim(x→∞) lnx/square root x
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 x0.5]:
limx→∞(lnx/x0.5) = (by l’Hôpital) limx→∞((x-1)/(0.5x-0.5))
which becomes
limx→∞2x-0.5 or limx→∞2/x0.5 = 0
By that reasoning, the reciprocal limit, limx→∞(x0.5/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.
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