Exercise 4.2.10 (Right and Left Limits). Introductory calculus courses typically refer to the right-hand limit of a function as the limit obtained by "letting x approach a from the right-hand side." (a) Give a proper definition in the style of Definition 4.2.1 for the right-hand and left-hand limit statements: lim f(x) = L and lim f(x) = M. x+a+ x-a (b) Prove that lima f(x) = L if and only if both the right and left-hand limits equal L. 116 Ve(L) L+€ L L-e Chapter 4. Functional Limits and Continuity ++ c-dc c + d Vs(c) Figure 4.4: DEFINITION OF FUNCTIONAL LIMIT. e-neighborhood Vε(L) centered at L, there is a point in the sequence-call it an-after which all of the terms an fall in Ve(L). Each e-neighborhood repre- sents a particular challenge, and each N is the respective response. For func- tional limit statements such as limc f(x) = L, the challenges are still made in the form of an arbitrary e-neighborhood around L, but the response this time is a d-neighborhood centered at c. - Definition 4.2.1 (Functional Limit). Let f AR, and let c be a limit point of the domain A. We say that limxc f(x) = L provided that, for all € > 0, there exists a > 0 such that whenever 0 < |x − c < 6 (and x = A) it follows that f(x) − L| < e. - This is often referred to as the "e-8 version" of the definition for functional limits. Recall that the statement |f(x) - L