TEXTBOOK NOTES - MODULE 2 LIMITS Demonstrate Understanding: Using complete sentences, in your own words, describe what is meant by the mathematical equation lim????????(????????) = ????????. (Not just how to read it, but your understanding of it.) ???????????????? Limits and Function Values: Given a function ????????(????????) and an ????????-value ????????, complete the following sentences to note the difference between ????????(????????) and lim????????(????????). ???????????????? 1. ????????(????????) is a real number and is the value... 2. lim????????(????????) is a real number and is the value... ???????????????? ONE-SIDED LIMITS Demonstrate Understanding: Using complete sentences, in your own words, describe what is meant by the following mathematical equations. (Not just how to read it, but your understanding of it.) 1. lim ????????(????????) = ???????? ????????????????+ 2. lim????????(????????) = ???????? ???????????????? Two-Sided Limits: Complete the following sentence to describe the connection between one-sided and two-sided limits. For some real number ????????, if lim????????(????????) ????????????????lim+????????(????????), then... ???????????????? DETERMINING LIMITS ALGEBRAICALLY Properties of Limits: Let ???????? and ???????? be two functions in which lim????????(????????) = ???????? and lim????????(????????) = ????????, where ????????, ???????????????? ???????????????? ????????, and ???????? are real numbers. Complete the equations below with the corresponding properties of limits. 1. lim???????? = ???????????????? 2. lim???????? = ???????????????? 3. lim[????????(????????) + ????????(????????)] = ???????????????? 4. lim[????????(????????) ????????(????????)] = ???????????????? 5. lim[????????????????(????????)] = ???????????????? 6. lim[????????(????????)????????(????????)] = ???????????????? ????????(????????) 7. lim = ????????????????????????(????????) 8. lim[????????(????????)]???????? = ???????????????? 9. lim ????????????????(????????) = ???????????????? Direct Substitution Property: Complete the following statement. If ???????? and ???????? are polynomials and ???????? is any real number, then... LIMITS AT INFINITY Demonstrate Understanding: Using complete sentences, in your own words, describe what is meant by the mathematical equation (Not just how to read it, but your understanding of it.) Show an Example: Sketch the graph of a function ????????(????????) that satisfies lim ????????(????????) = 0 and lim ????????(????????) = 3. ???????? ???????? 1 Theorem 1.3: If ???????? is a positive integer, then lim ???????? ???????? = ???????? INFINITE LIMITS Show an Example: Sketch the graph of a function ????????(????????) that satisfies lim????????(????????) = and ????????1lim+????????(????????) = . ????????1