66 3.1 Limits of Functions (ii) limx →x0 f (x)= limx →x0 h(x)= ℓ. Then limx →x0 g(x)= ℓ. Proof: The proof is straightforward using Theorem 2.1.3 and Theorem 3.1.2. □ Remark 3.1.3 We will adopt the following convention. When we write limx →x0 f (x) without specifying the domain D of f we will assume that D is the largest subset of R such that if x ∈ D, then f (x) results in a real number. For example, in 1 lim x→2 x + 3 we assume D = R\{−3} and in √ lim x x→1 we assume D =[0,∞). Exercises 3.1.1 Use the defnition of limit to prove that (a) limx →2(3x − 7)= −1. (b) limx →3(x2 + 1)= 10. x + 3 (c) lim = 2. x→1 x + 1 √ (d) limx →0 x = 0. (e) limx →2 x3 = 8. 3.1.2 Let f : D → R and let x0 be a limit point of D. Prove that if f has a limit at x0, then this limit is unique. (Hint: the argument is analogous to the one used in the proof of Theorem 2.1.1.) 3.1.3 Prove Proposition 3.1.1. 3.1.4 Let I =(a,b) for a,b ∈ R, a < b. Prove that if c ∈ I, then c is a limit point of I \{c}. 3.1.5 Prove Corollary 3.1.3 3.1.6 Prove Corollary 3.1.4 3.1.7 Using Corollary 3.1.4, prove that the following limits do not exist. x (a) lim . x→0 |x| (b) lim sin(1/x). x→0 3.1.8 Let f : D → R and let x0 be a limit point of D. Prove that if limx →x0 f (x)= ℓ, then lim | f (x)| = |ℓ|. x→x0 Give an example to show that the converse is not true in general.
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