70 3.2 Limit Theorems Defnition 3.2.2 Given a subset D of R, we say that x0 is a left limit point of D if for every δ > 0, the open interval (x0 − δ ,x0) contains a point x of D, x ̸ = x0. Similarly, x0 is called a right limit point of D if for every δ > 0, the open interval (x0,x0 + δ ) contains a point x of D, x ̸ = x0. It follows from the defnition that x0 is a limit point of D if and only if it is a left limit point of D or it is a right limit point of D. Defnition 3.2.3 (One-sided limits) Let f : D → R and 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 − f (x)= ℓ. x→x0 The right-hand limit of f at x0 can be defned in a similar way and is denoted lim + f (x). x →x0 ■ Example 3.2.4 Consider the function f : R \{0}→ R given by f (x)= |x|/x. Let x0 = 0. Note frst that 0 is a limit point of the set D = R \{0}. For x > 0, we have f (x)= x/x = 1 and therefore lim + f (x)= limx →0+ 1 = 1. x →x0 Similarly, for x < 0 we have f (x)= −x/x = −1. Therefore, lim − f (x)= limx →0−− 1 = −1. x →x0 ■ Example 3.2.5 Consider the function f : R → R given by ( 2 − 1, x if x ≥−1; f (x)= (3.2) x + 4, if x < −1. We have 2 − 1)= 0, lim f (x)= lim (x x→−1+ x→−1+ and lim f (x)= lim (x + 4)= 3, x→−1− x→−1− The following theorem follows directly from the defnition of one-sided limits. The proof is left as an exercise. Theorem 3.2.3 Let f : D → R and let x0 be both a left limit point of D and a right limit point of D. Then the following are equivalent: (i) lim f (x)= ℓ. x→x0 (ii) lim f (x)= ℓ = lim f (x). + − x→x x→x 0 0 |x| ■ Example 3.2.6 It follows from Example 3.2.4 that lim does not exist, since the one-sided x→0 x limits do not agree.
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