Mark M. answered • 10/08/20
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Retired math prof. Calc 1, 2 and AP Calculus tutoring experience.
As x → 0+, lnx → -∞ and x → 0. So the limit has the indeterminate form (0)(-∞).
Rewriting the limit as limx→0+ [lnx / (1/x)] we have the indeterminate form -∞/∞.
Applying L'Hopital's Rule, the given limit is equivalent to limx→0+ [(1/x) / (-1/x2)] =
limx→0+(-x) = 0
