Mathematics Foundations/4.4 Limits and Convergence

Limits and Convergence Introduction to Limits The concept of limits is fundamental to calculus and mathematical analysis. While we introduced limits of sequences in the previous section, here we extend the notion to functions. Limits of Functions Informal Definition Intuitively, we say that if the values of get arbitrarily close to as gets arbitrarily close to (but not equal to ). Formal Definition (Epsilon-Delta) Let be a function defined on some open interval containing , except possibly at itself. We say that if for every , there exists a such that: This definition is known as the epsilon-delta definition of a limit. One-Sided Limits The left-hand limit is denoted and means that approaches as approaches from values less than . The right-hand limit is denoted and means that approaches as approaches from values greater than . Theorem: if and only if both and . Properties of Limits If and , then: 1. Sum Rule: 2. Difference Rule: 3. Product Rule: 4. Quotient Rule: provided 5. Constant Multiple Rule: for any constant 6. Power Rule: for any integer 7. Root Rule: for any integer , provided that when is even, Limits at Infinity We can also consider limits as approaches infinity. Definition We say if for every , there exists a positive number such that: Similarly, if for every , there exists a negative number such that: Examples Infinite Limits A function can also have an infinite limit. Definition We say if for every positive number , there exists a such that: Similarly, if for every negative number , there exists a such that: Examples Continuity Continuity is a fundamental property of functions that is defined using limits. Definition A function is continuous at a point if the following three conditions are satisfied: 1. is defined 2. exists 3. A function is continuous on an interval if it is continuous at every point in the interval. Types of Discontinuities 1. Removable discontinuity: exists but either is not defined or 2. Jump discontinuity: Both one-sided limits exist but are not equal 3. Infinite discontinuity: At least one of the one-sided limits is infinite 4. Oscillatory discontinuity: The limit does not exist due to oscillation Properties of Continuous Functions If and are continuous at , then: 1. is continuous at 2. is continuous at 3. is continuous at 4. is continuous at provided 5. (the composition) is continuous at Important Theorems on Continuity Intermediate Value Theorem Theorem: If is continuous on the closed interval and , then for any value between and , there exists at least one point such that . This theorem guarantees that a continuous function cannot "jump" over any value between its function values. Extreme Value Theorem Theorem: If is continuous on a closed interval , then attains both a maximum and a minimum value on . In other words, there exist points such that for all . Uniform Continuity Uniform continuity is a stronger notion than continuity. Definition A function is uniformly continuous on a set if for every , there exists a such that for all : The key difference from ordinary continuity is that depends only on and not on the point in the domain. Heine-Cantor Theorem Theorem: If is continuous on a closed interval , then is uniformly continuous on . Limits of Sequences Revisited There is a deep connection between limits of functions and limits of sequences. Relationship Between Sequential and Functional Limits Theorem: Let be defined on a neighborhood of , except possibly at itself. Then if and only if for every sequence such that for all and , we have . This theorem provides an alternative characterization of limits using sequences. L'Hôpital's Rule L'Hôpital's Rule is a powerful tool for evaluating limits that yield indeterminate forms. Statement of L'Hôpital's Rule Theorem: Suppose and are differentiable on an open interval containing (except possibly at ), and for all with . If or and , and exists (or equals ), then: This rule applies to the indeterminate forms and . Other Indeterminate Forms L'Hôpital's Rule can be extended to handle other indeterminate forms: These forms can often be rewritten as or using algebraic manipulations or logarithms. Convergence of Functions Beyond pointwise limits, we can consider how functions converge to other functions. Pointwise Convergence A sequence of functions converges pointwise to a function on a set if for every : Uniform Convergence A sequence of functions converges uniformly to a function on a set if for every , there exists an such that for all and all : Uniform convergence is stronger than pointwise convergence and preserves important properties like continuity. Weierstrass M-Test Theorem: Let be a sequence of functions defined on a set . If there exists a sequence of positive numbers such that for all and converges, then converges uniformly on . This test is particularly useful for determining the uniform convergence of series of functions. Exercises 1. Evaluate using the definition of a limit. 2. Find . 3. Determine all points of discontinuity of the function and classify them. 4. Use the Intermediate Value Theorem to show that the equation has at least one real solution. 5. Evaluate using L'Hôpital's Rule. 6. Determine whether the sequence of functions converges uniformly on . 7. Prove that if is uniformly continuous on , then can be extended to a continuous function on . 8. Use the Weierstrass M-Test to show that the series converges uniformly on .