Presentation 1 Limits from Tables and Graphs PDF

Limits from Tables and Graphs Basic Calculus  At the end of the module, students should Learning be able to: Objectives  Illustrate the limit of a function using table of values...

Limits from Tables and Graphs Basic Calculus  At the end of the module, students should Learning be able to: Objectives  Illustrate the limit of a function using table of values and graph of the function. 2nd Term 2024-2025 Limits from Tables and Graphs 2 Introduction Sir Isaac Newton, an English physicist and mathematician, and Gottfried Wilhelm Leibniz, a German philosopher and mathematician, independently developed the foundations of calculus in the mid-17th century. Calculus has widespread applications in Science, Engineering, as well as in Economics. It is a study of limits, functions, derivatives, integrals, and infinite series. 2nd Term 2024-2025 Limits from Tables and Graphs 3 Limits Defining Limits The limit of a function f(x) is the value it approaches as the value of x approaches a certain value. “As x approaches a, the limit of f(x) approaches L”. This is written in symbols as follows: 2nd Term 2024-2025 Limits from Tables and Graphs 5  Example 1 Find. Solution The arrow pointing at 4 indicates that x is approaching 4 from the left side and from the right side. This means that x can take on values less than 4 and values greater than 4. It cannot take 4 as a value because it is just approaching 4. The first thing to do to find the limit of the given function is to construct the table of values. In table 1, x takes on some values of x that are less than 4. These values of x get closer and closer to 4. In table 2, x takes on some values that are greater than 4. These values of x get closer and closer to 4. 2nd Term 2024-2025 Limits from Tables and Graphs 6 2nd Term 2024-2025 Limits from Tables and Graphs 7 2nd Term 2024-2025 Limits from Tables and Graphs 8 As the value of x gets closer to 4 from the left or as x approaches 4 from left, the value of f(x) approaches 6. As the value of x gets closer and closer to 4 from the right or as x approaches 4 from the right, f(x) approaches 6. In other words, the value of f(x) gets closer and closer to 6 as the value of x gets closer and closer to 4 from either side. This can be written as follows: This means that the limit of (x + 2) is 6 as x approaches 4 from either side. 2nd Term 2024-2025 Limits from Tables and Graphs 9  Example 2 Find. Solution Let f(x) =. Notice that f(x) is not defined at x = 3. If x is 3, then f(x) = which is undefined. In the language of calculus, it is indeterminate. The expression should be simplified by factoring its numerator. This can be done because x ≠ 3; it is only approaching 3. = = 2nd Term 2024-2025 Limits from Tables and Graphs 10 2nd Term 2024-2025 Limits from Tables and Graphs 11 2nd Term 2024-2025 Limits from Tables and Graphs 12 As the value of x that is less than 3 gets closer and closer to 3, the value of f(x) gets closer and closer to 2. As the value of x that is greater than 3 gets closer and closer to 3, the value of f(x) gets closer and closer to 2. Hence, = = = 2. 2nd Term 2024-2025 Limits from Tables and Graphs 13  Example 3 Find. Solution Let f(x) =. Notice that f(x) is not defined when x = 1. Factor the numerator of the expression and cancel one of the factors with (x – 1). = = 2nd Term 2022-2023 Limits from Tables and Graphs 14 2nd Term 2024-2025 Limits from Tables and Graphs 15 2nd Term 2024-2025 Limits from Tables and Graphs 16 As the value of x gets closer and closer to 1 from the left, or as x approaches 1 from the left, the value of f(x) approaches 4. As the value of gets closer and closer to 1 from the right or as x approaches 1 from the right, f(x) approaches 4. In other words, the value of f(x) gets closer and closer to 4 as the value of x gets closer and closer to 1 from either side. Hence, = = = 4. 2nd Term 2024-2025 Limits from Tables and Graphs 17 One-Sided Limits  Example 4 Let f be defined by the equation f(x) =. a. Evaluate b. Evaluate Solution c. The = does not exist because is not a real number if x > 3. d. Construct the table of values. 2nd Term 2024-2025 Limits from Tables and Graphs 19 2nd Term 2024-2025 Limits from Tables and Graphs 20 2nd Term 2024-2025 Limits from Tables and Graphs 21  When a Limit Does Not Exist  Example 5 When a Limit Does Not Exist Solution The graph is shown on the next page. 2nd Term 2024-2025 Limits from Tables and Graphs 23 2nd Term 2024-2025 Limits from Tables and Graphs 24  a. = -2 b. = 3x = 3(0) = 0 c. ≠ Note that when ≠ , the does not exist. Hence, the does not exist. 2nd Term 2024-2025 Limits from Tables and Graphs 25  Activity #1 a. b. c. 2nd Term 2024-2025 Limits from Tables and Graphs 26  Windel C. Austria Thank you [email protected] 2nd Term 2024-2025 Limits from Tables and Graphs 27