Introduction to Mathematical Analysis I - Second Edition

70 3.2 LIMIT THEOREMS It follows from the definition that a is a limit point of A if and only if it is a left limit point of A or it is a right limit point of A . Definition 3.2.3 (One-sided limits) Let f : D → R and let ¯ x be a left limit point of D . We write lim x → ¯ x − f ( x ) = ` if for every ε > 0, there exists δ > 0 such that | f ( x ) − ` | < ε for all x ∈ B − ( ¯ x ; δ ) . We say that ` is the left-hand limit of f at ¯ x . The right-hand limit of f at ¯ x can be defined in a similar way and is denoted lim x → ¯ x + f ( x ) . Example 3.2.4 Consider the function f : R \ { 0 } → R given by f ( x ) = | x | x . Let ¯ x = 0. Note first that 0 is a limit point of the set D = R \ { 0 } → R . Since, for x > 0, we have f ( x ) = x / x = 1, we have lim x → ¯ x + f ( x ) = lim x → 0 + 1 = 1 . Similarly, for x < 0 we have f ( x ) = − x / x = − 1. Therefore, lim x → ¯ x − f ( x ) = lim x → 0 − − 1 = − 1 . Example 3.2.5 Consider the function f : R → R given by f ( x ) = ( x + 4 , if x < − 1; x 2 − 1 , if x ≥ − 1 . (3.4) We have lim x →− 1 + f ( x ) = lim x →− 1 + x 2 − 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. Theorem 3.2.3 Let f : D → R and let ¯ x be both a left limit point of D and a right limit point of D . Then lim x → ¯ x f ( x ) = ` if and only if lim x → ¯ x + f ( x ) = ` and lim x → ¯ x − f ( x ) = `.