104 4.4 L’Hôpital’s Rule Exercises 4.3.1 (a) Let f : R → R be differentiable. Prove that if f ′ (x) is bounded, then f is Lipschitz continuous and, in particular, uniformly continuous. (b) Give an example of a function f : (0,∞) → R which is differentiable and uniformly continuous but such that f ′ (x) is not bounded. 4.3.2 ▶ Let f : R → R. Suppose there exist ℓ ≥ 0 and α > 0 such that | f (u) − f (v)|≤ ℓ|u − v| α for all u,v ∈ R. (4.7) (a) Prove that f is uniformly continuous on R. (b) Prove that if α > 1, then f is a constant function. (c) Find a nondifferentiable function that satisfes the condition above for α = 1. 4.3.3 ▷ Let f and g be differentiable functions on R such that f (x0)= g(x0) and f ′ (x) ≤ g ′ (x) for all x ≥ x0. Prove that f (x) ≤ g(x) for all x ≥ x0. 4.3.4 Let f,g: R → R be differentiable functions satisfying (a) f (0)= g(0)= 1 f ′ (x) g ′ (x) (b) f (x) > 0, g(x) > 0 and > for all x. f (x) g(x) Prove that f (1) g(1) > 1 > . g(1) f (1) 4.3.5 ▷ Let f be twice differentiable on an open interval I. Suppose that there exist a,b,c ∈ I with a < b < c such that f (a) < f (b) and f (b) > f (c). Prove that there exists d ∈ (a,c) such that f ′′ (d) < 0. 4.3.6 ▷ Prove that the function f defned in Exercise 4.1.11 is not monotone on any open interval containing 0. 4.4 L’Hôpital’s Rule We now prove a result that allows us to compute various limits by calculating a related limit involving derivatives. All four theorems in this section are known as L’Hôpital’s Rule. In this section, we assume a,b ∈ R with a < b. Theorem 4.4.1 Let f and g be continuous on [a,b] and differentiable on (a,b). Suppose that: (i) f (x0)= g(x0)= 0, where x0 ∈ [a,b], (ii) there exists δ > 0 such that g ′ (x)≠ 0 for all x ∈ (x0 − δ ,x0 + δ ) ∩ [a,b], x ≠ x0.
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