1. Calculus and Analytic Geometry I Cloud County Community College Fall, 2012 Instructor: Timothy L. Warkentin

2. Chapter 02: Limits and Continuity 02.01 Rates of Change and Tangents to Curves 02.02 Limit of a Function and Limit Laws 02.03 The Precise Definition of a Limit 02.04 One-Sided Limits 02.05 Continuity 02.06 Limits Involving Infinity; Asymptotes of Graphs

3. Chapter 02 Overview The formal definition of the Limit of a Function is one of the most intellectually satisfying in all of mathematics. It depends on the idea that a test is needed to determine if two quantities are equal. This test says that two quantities will be equal if their difference can be shown to be less than any amount whatsoever. This concept of a limit allowed Calculus (derivatives and definite integrals) to be established on a mathematically rigorous foundation and resolved Zeno's paradoxes. The sections in this chapter will be presented in the following order: 02.01, 02.03, 02.02, 02.04, 02.05, 02.06.

4. 02.01: Rates of Change and Tangents to Curves 1 The Average Rate of Change (ARC) computed as the average of the changes between points in a discrete data set and as the slope of the secant line between the first and last data points. Example data on next slide The ARC using the slope of the secant line. Examples 1 & 4 Finding the Instantaneous Rate of Change (IRC) of a continuous function using the Difference Quotient (DQ). Examples 2 & 5 The limit of the DQ and the slope of the tangent line. Example 3

5. 02.01: Rates of Change and Tangents to Curves 2 DatePopulationPopulation Change 19001,470,4950 19101,690,949+220,454 19201,769,257+78,308 19301,880,999+111,742 19401,801,028-79,971 19501,905,299+104,271 19602,178,611+273,312 19702,246,578+67,967 19802,363,679+117,101 19902,477,574+113,895 20002,688,418+210,844

6. 02.02: Limit of a Function and Limit Laws 1 Limits laws are used to avoid epsilon/delta proofs. Example 5 1. 2. 3. 4. 5.

7. 02.02: Limit of a Function and Limit Laws 2 Many limits may be found by substitution. Calculus operations are straightforward when dealing with polynomial functions. If P[x] is a polynomial then (theorem 2, page 69). Example 6 When division by zero prevents the limit of a function from being taken by using substitution the form of the function must be changed. Examples 7 & 9 Using software to estimate limits. Example 8

8. 02.02: Limit of a Function and Limit Laws 3 The Sandwich theorem sometimes allows very difficult limits (the meat) to be found. The problem is finding the bread. Examples 10 & 11 Two important limits and another limit law found using the Sandwich theorem. 1.2. 6.

9. 02.03: The Precise Definition of a Limit 1 Understanding absolute value notation. Examples 1 & 4 Multiplication of an inequality by a negative reverses the inequality.

10. 02.03: The Precise Definition of a Limit 2 The Precise Definition of a Limit is a marvel of mathematical engineering. Let f [x] be defined on an open interval about x 0 except possibly at x 0 itself. Then If, for every ε > 0, there exists a δ > 0 such that Example 2

11. 02.04: One-Sided Limits 1 The limits discussed in the previous sections are actually two-sided limits where the function must have the same limiting value on either side of x 0. The absolute value notation in the limit definition is a compact version of Examples 1 & 2 The epsilon/delta definitions given in the text for right and left hand limits are precisely the two definitions implied by the absolute value notation in the epsilon/delta definition of a limit given in section 3.

12. 02.04: One-Sided Limits 2 Important limits: Examples 4, 5, & 6 1. 2. 3. 4.

13. 02.05: Continuity 1 Continuity at a Point Examples 1, 2 & 3 –Interior Point: A function y = f [x] is continuous at an interior point c of its domain if –Endpoint: A function y = f [x] is right/left continuous at a left/right endpoint a/b of its domain if

14. 02.05: Continuity 2 A function is left/right continuous if the function is continuous at all points when approached from the left/right. Example 4 A function is continuous on an interval if and only if (iff) it is continuous at every point in the interval. Example 5 The continuity laws (theorem 8, page 95) are analogous to the limit laws since continuity depends on the existence of limits. Examples 6 & 7 The inverse function of any continuous function is continuous over its domain. The composition of continuous functions is continuous. Example 8

15. 02.05: Continuity 3 Another limit law dealing with the composition of functions can be established using continuity. Example 9 7. Types of Discontinuities –Jump (also called Step) –Asymptotic (also called Infinite, Essential) (look like slip faults or volcanos) –Removable (also called Point, Hole): The domain can be extended by use of a piecewise definition so that the function is continuous. This process is called “Continuous Extension to a Point”. Example 10

16. 02.05: Continuity 4 The Intermediate Value Theorem for Continuous Functions Examples 11 & 12 –If is continuous on and then there is at least one number such that.

17. 02.06: Limits Involving Infinity; Asymptotes of Graphs 1 The symbol for infinity (∞) does not represent a Real number. Rather it is meant to convey the idea of an amount beyond all finite (large or small) bounds. Thus the limit of a function at either plus or minus infinity is not like an ordinary limit where the limit occurs at a specific point in the domain. The precise definition of a limit at infinity rests upon the idea that two quantities are equal when they are less different by an amount (epsilon > 0) whatsoever. It is very similar to the definition used to define the sum of infinite series. The limit laws at infinity are the same as those already presented. Examples 2 & 3

18. 02.06: Limits Involving Infinity; Asymptotes of Graphs 2 Limits at infinity are used to provide a definition for horizontal asymptotes. Examples 4 – 9 * The line y = b is a horizontal asymptote of f[x] if either Rational functions may have horizontal and/or oblique asymptotes. Asymptotes are found using long division to produce the form The quotient q[x] is the required asymptote. Example 10

19. 02.06: Limits Involving Infinity; Asymptotes of Graphs 3 Limits at Infinity vs. Infinite Limits. When the value of a function approaches plus or minus infinity at some domain value the limit is not a normal limit because the value approached is not a Real number. Examples 11, 12 & 13 Infinite limits provide a precise definition for vertical asymptotes. Examples 15 – 18 * The line x = a is a vertical asymptote of f[x] if either Dominant terms and the far left/right behavior of functions. Example 19