1 Comparing Growth of ln(x) and x 3 We have one more item on our original list of limits to cover; again we’ll look at a slight variation on the original problem. We’re going to ﬁnd: lim x→∞ This limit is of the form ln x x→∞ x1/3 lim ∞ ∞, ln x . x1/3 so we apply l’Hôpital’s rule to ﬁnd: = = = lim 1/x x→∞ 1 x−2/3 3 lim 3x−1/3 x→∞ (l’Hop) 0 We conclude that ln x grows more slowly as x approaches inﬁnity than x1/3 or any positive power of x. In other words, ln x increases very slowly. Question: When we discussed extensions of l’Hôpital’s rule, we learned that we’re allowed to change some hypotheses. How many hypotheses can we change at once? f (a) Answer: We can make any or all of the three changes listed. However, g(a) ∞ 0 , − , or . must always be of the form ∞ ∞ ∞ 0

Comparing Growth of ln(x) and x^1/3

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