Mathematics Foundations/9.1 Limits and Continuity

9.1 Limits and Continuity Introduction Limits and continuity form the foundation of calculus. These concepts allow us to analyze the behavior of functions as their inputs approach specific values, even when the function may not be defined at those points. Intuitive Understanding of Limits The limit of a function as approaches a value represents the value that gets arbitrarily close to as gets arbitrarily close to , but not equal to . We denote this as: Example 1 Consider the function . This function is undefined at , but we can analyze its limit as approaches 1: Formal Definition of Limits The formal definition of a limit, known as the epsilon-delta definition, provides a rigorous mathematical foundation: A function has a limit as approaches if for every , there exists a such that: Visualizing the Epsilon-Delta Definition The epsilon-delta definition can be understood geometrically: defines a horizontal band around with width defines a vertical band around with width The limit exists if we can always find a suitable for any given Properties of Limits If the limits and exist, 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 (assuming when ) One-Sided Limits Sometimes a function approaches different values from the left and right sides of a point. Left-hand limit: (as approaches from values less than ) Right-hand limit: (as approaches from values greater than ) The limit exists if and only if both one-sided limits exist and are equal: if and only if Limits at Infinity We can also examine the behavior of functions as approaches positive or negative infinity: means approaches as grows arbitrarily large means approaches as becomes arbitrarily large negative Example 2 Continuity A function is continuous at a point if all three of these conditions are met: 1. is defined 2. exists 3. In other words, a function is continuous at a point if its limit at that point equals its function value. Types of Discontinuities 1. Removable discontinuity: The limit exists but either the function is undefined at that point or the function value doesn't equal the limit 2. Jump discontinuity: The left and right limits exist but are not equal 3. Infinite discontinuity: The limit approaches positive or negative infinity 4. Oscillatory discontinuity: The function oscillates infinitely as it approaches the point Continuity on Intervals A function is continuous on an open interval if it is continuous at every point in the interval A function is continuous on a closed interval if it is continuous on and: * (right-continuous at ) * (left-continuous at ) Important Theorems on Continuity Intermediate Value Theorem If a function is continuous on a closed interval and , then for any value between and , there exists at least one point such that . Extreme Value Theorem If a function is continuous on a closed interval , then attains both a maximum and minimum value on that interval. Uniform Continuity A function is uniformly continuous on an interval if for every , there exists a such that for all : Every continuous function on a closed, bounded interval is uniformly continuous. Exercises 1. Evaluate . 2. Determine where the function is continuous. 3. Prove that does not exist. 4. Show that the function for and is continuous at . 5. Apply the Intermediate Value Theorem to show that the equation has at least one real solution.