Lecture Notes: Limits and Continuity 1 Limits 1.1 Introduction to Limits The concept of a limit is fundamental to calculus. It describes the behavior of a function as its input approaches a particular value. 1.1.1 Informal Definition If f (x) becomes arbitrarily close to a single number L as x approaches c (from either side), then we write: lim f (x) = L x→c and say ”the limit of f (x) as x → c is L.” 1.2 The Formal Definition of a Limit The precise definition of a limit (using ϵ-δ) is as follows: Let f (x) be defined on an open interval containing c, except possibly at c. Then: lim f (x) = L x→c if for every ϵ > 0, there exists a δ > 0 such that: 0 < |x − c| < δ =⇒ |f (x) − L| < ϵ. 1.2.1 Key Terms • ϵ: A measure of how close f (x) is to L. • δ: A measure of how close x is to c. 1 1.3 One-Sided Limits If x approaches c from the left (x → c− ) or the right (x → c+ ), the limits are called one-sided: lim− f (x) and lim+ f (x). x→c x→c A two-sided limit exists if and only if: lim f (x) = lim+ f (x) = L. x→c− 1.4 1.4.1 x→c Properties of Limits Basic Limit Laws Let limx→c f (x) = L and limx→c g(x) = M . Then: 1. Sum Rule: limx→c [f (x) + g(x)] = L + M . 2. Difference Rule: limx→c [f (x) − g(x)] = L − M . 3. Product Rule: limx→c [f (x)g(x)] = L · M . 4. Quotient Rule: limx→c f (x) g(x) = L , M provided M ̸= 0. 5. Constant Multiple Rule: limx→c [k·f (x)] = k·L, where k is constant. 1.4.2 Squeeze Theorem If g(x) ≤ f (x) ≤ h(x) for all x in some interval around c, and: lim g(x) = lim h(x) = L, x→c x→c then: lim f (x) = L. x→c 1.5 Special Limits 1. Limits Involving Infinity: 1 lim = 0, x→∞ x lim x→−∞ 1 = 0. x 2. Exponential Limits: lim (1 + x)1/x = e. x→0 2 1.6 1.6.1 Examples of Limits Example 1: A Simple Limit Evaluate limx→2 (3x2 − 4): Solution: Substitute x = 2. lim (3x2 − 4) = 3(22 ) − 4 = 12 − 4 = 8. x→2 1.6.2 Example 2: A Limit Using the Squeeze Theorem Evaluate limx→0 x2 sin x1 : 1 2 2 Solution: − x ≤ x sin ≤ x2 . x Using the Squeeze Theorem: lim −x2 = 0 and x→0 Therefore: Continuity 2.1 x→0 1 lim x sin = 0. x→0 x 2 2 lim x2 = 0. Definition of Continuity A function f (x) is continuous at c if: 1. f (c) is defined. 2. limx→c f (x) exists. 3. limx→c f (x) = f (c). 3 2.2 Types of Discontinuity 1. Removable Discontinuity: The limit exists, but f (c) is not defined or f (c) ̸= limx→c f (x). 2. Jump Discontinuity: The left-hand and right-hand limits exist but are not equal. 3. Infinite Discontinuity: The function approaches ±∞ as x → c. 2.3 Intermediate Value Theorem (IVT) If f (x) is continuous on [a, b], and N is a value between f (a) and f (b), then there exists c ∈ (a, b) such that f (c) = N . 2.4 2.4.1 Examples of Continuity Example 1: Checking Continuity Determine if f (x) = x2 −1 x−1 is continuous at x = 1: Solution: Simplify f (x) as f (x) = x + 1, for x ̸= 1. The function is not defined at x = 1, so it has a removable discontinuity. 2.4.2 Example 2: Applying the IVT Show that f (x) = x3 − x − 1 has a root in [1, 2]: Solution: Compute f (1) = −1 and f (2) = 5. Since f (x) is continuous on [1, 2] and 0 ∈ [−1, 5], there exists c ∈ (1, 2) such that f (c) = 0. 4

Limits and Continuity Lecture Notes

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