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 Dof f we will assume that Dis the largest subset of Rsuch that if x ∈D, then f (x) results in a real number. For example, in lim x→2 1 x+3 we assume D=R\{−3} and in lim x→1 √x we assume D=[0,∞). Exercises 3.1.1 Use the definition of limit to prove that (a) limx →2(3x−7)=−1. (b) limx →3(x2 +1)=10. (c) lim x→1 x+3 x+1 =2. (d) limx →0 √x =0. (e) limx →2 x3 =8. 3.1.2 Let f : D→Rand 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, thenc 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. (a) lim x→0 x |x| . (b) lim x→0 sin(1/x). 3.1.8 Let f : D→Rand let x0 be a limit point of D. Prove that if limx →x0 f (x)=ℓ, then lim x→x0 | f (x)| =|ℓ|. Give an example to show that the converse is not true in general.
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