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(x 2 +1) =10. (c) lim x→1 x+3 x+1 =2. (d) limx→0√x =0. (e) limx→2x 3 =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, 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. (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.
RkJQdWJsaXNoZXIy NTc4NTAz