109 ■ Example 4.4.6 Consider the limit lim x→∞ 1 x( π 2 −arctanx) . Writing the quotient in the form 1/x π 2 −arctanx , we can apply Theorem 4.4.3. We now compute the limit of the quotient of the derivatives lim x→∞ −(1/x2) − 1 x2+1 =lim x→∞ x2 +1 x2 =1. In view of Theorem 4.4.3 the desired limit is also 1. The following theorem can be proved following the method in the proof of Theorem 4.4.2. Theorem 4.4.4 Let f andg be differentiable on(a,∞). Suppose that: (i) g′(x)̸=0 for all x ∈(a,∞), (ii) limx →∞ f (x)=limx →∞ g(x)=∞, (iii) limx →∞ f ′(x) g′(x) =ℓ, for some ℓ ∈R. Then lim x→∞ f (x) g(x) =ℓ. ■ Example 4.4.7 Consider the limit lim x→∞ lnx x . Clearly the functions f (x)=lnx andg(x)=x satisfy the conditions of Theorem 4.4.4. We have lim x→∞ f ′(x) g′(x) =lim x→∞ 1/x 1 =0 It follows from Theorem 4.4.4 that limx →∞ lnx x =0. Exercises 4.4.1 Use L’Hôpital’s rule to find the following limits (you may assume known all the relevant derivatives from calculus): (a) lim x→−2 x3 −4x 3x2 +5x−2 . (b) lim x→0 ex −e−x sinxcosx .
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