Introduction to Mathematical Analysis I - 3rd Edition

70 3.2 Limit Theorems Definition 3.2.2 Given a subset Dof R, we say that x0 is a left limit point of Dif for everyδ >0, the open interval (x0−δ,x0) contains a point x of D, x̸=x0. Similarly, x0 is called a right limit point of Dif for every δ >0, the open interval (x0,x0 +δ) contains a point x of D, x̸ =x0. It follows from the definition that x0 is a limit point of Dif and only if it is a left limit point of D or it is a right limit point of D. Definition 3.2.3 (One-sided limits) Let f : D→Rand let x0 be a left limit point of D. We say that f has a left-hand limit at x0 if there exists a real number ℓ such that for any real number ε >0, there exists δ >0 such | f (x)−ℓ| <ε for all x ∈(x0 −δ,x0). In this case, we say that ℓ is the left-hand limit of f at x0 and write lim x→x− 0 f (x) =ℓ. The right-hand limit of f at x0 can be defined in a similar way and is denoted limx →x+ 0 f (x). ■ Example 3.2.4 Consider the function f : R\ {0} →Rgiven by f (x) =|x|/x. Let x0 =0. Note first that 0 is a limit point of the set D=R\ {0}. For x >0, we have f (x) =x/x =1 and therefore limx →x+ 0 f (x) =limx→0+1=1. Similarly, for x <0 we have f (x) =−x/x =−1. Therefore, limx →x− 0 f (x) =limx→0−−1=−1. ■ Example 3.2.5 Consider the function f : R→Rgiven by f (x) =( x2 −1, if x ≥ −1; x+4, if x <−1. (3.2) We have lim x→−1+ f (x) = lim x→−1+ (x2 −1) =0, and lim x→−1− f (x) = lim x→−1− (x+4) =3, The following theorem follows directly from the definition of one-sided limits. The proof is left as an exercise. Theorem 3.2.3 Let f : D→Rand let x0 be both a left limit point of Dand a right limit point of D. Then the following are equivalent: (i) lim x→x0 f (x) =ℓ. (ii) lim x→x+ 0 f (x) =ℓ = lim x→x− 0 f (x). ■ Example 3.2.6 It follows from Example 3.2.4 that lim x→0 |x| x does not exist, since the one-sided limits do not agree.

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