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Unit A Overview

Unit A Overview. Limits and Continuity. Limits. Definition of a limit. The limit is a method of evaluating an expression as an argument approaches a value. This value can be any point on the number line and often limits are evaluated as an argument approaches infinity or negative infinity.

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Unit A Overview

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  1. Unit A Overview Limits and Continuity

  2. Limits

  3. Definition of a limit • The limit is a method of evaluating an expression as an argument approaches a value. This value can be any point on the number line and often limits are evaluated as an argument approaches infinity or negative infinity.

  4. Limit Notation • The following expression states that as x approaches the value c the function approaches the value L.

  5. Properties of limits • Sum Rule • The limit of the sum of 2 functions is the sum of their limits • Difference Rule • The limit of the difference of 2 functions is the difference of their limits • Product Rule • The limit of a product of 2 functions is the product of their limits

  6. Properties of limits • Constant Multiple Rule • The limit of a constant times a function is the constant times the limit of the function • Quotient Rule • The limit of a quotient of two functions is the quotient of their limits, provided that the denominator is not zero. • Power Rule • If r and s are integers and s ≠ 0, then

  7. Examples =19 =4(2)²+3 = =

  8. ALWAYS FACTOR FIRST If you plug 3 directly in, you will get an indeterminate form. Factor first then evaluate. f(x) = f(x) = = f(x) =

  9. Finding Limits Graphically • When finding a limit graphically you look at either side of the value you are wanting, just like you did on a table.

  10. Finding limits graphically continued • It does not matter what is actually happening at the point in question, just on either side. =6

  11. Practice Problem Finding Limits Graphically Find 1

  12. One and Two Sided Limits • Sometimes the values of a function f tend to different limits as x approaches a number c from the left and from the right. • Right-hand limit: • The limit of f as x approaches c from the right. • Left-hand limit: • The limit of f as x approaches c from the left

  13. Limits that Do Not Exist • A function f(x) has a limit as x approaches c if and only if (IFF) the right-hand and left-hand limits at c exist and are equal. • If those limits are not equal, it is said that the limit does not exist. DNE.

  14. Exploring left and right hand limits F(2) = 2 = 2 F(1) = 2 = 1 = 3 = 1 = DNE = 1 = DNE F(4)

  15. Piecewise Functions • Sketch the graph of f. Then identify the values for c for which f(x) = { 3x+1, x<0 { x, 0≤ x≤ 2 { 2, x > 2 at all points on the graph except when c = 0

  16. Limits involving Infinity • Let f be the function f(x) = • As you can see • If f(x) approaches ± ∞ as x approaches c from the right or left then the line x =c is a vertical asymptote of the graph. -∞ ∞

  17. Limits Approaching Infinity • Horizontal Asymptote • The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either or

  18. Limits Approaching Infinity 2 • = • = • Therefore the function has a horizontal asymptote at y = 2 2

  19. End Behavior Models • For numerically large values of x, we can sometimes model the behavior of a complicated function by a simpler one that acts virtually the same way.

  20. Example • Let f(x) = - + 3x² - 5x + 6 and g(x) = • While f and g are quite different for numerically small values, they are virtually identical for large x values.

  21. Limits approaching infinity analytically • End behavior model is used when evaluating a polynomial and rational function as it approaches infinity. • Example • If you plugged in a very large number you would get a fraction that would reduce to almost the same thing every time.

  22. Limits approaching infinity analytically • Take the end behavior model for top and bottom and then reduce. = = making it the end behavior model

  23. Limits approaching infinity analytically = = making it the end behavior model

  24. Using End Behavior do Identify horizontal asymptote • Notice in the example below, y=2/3 is also the horizontal asymptote of the graph. • Take the end behavior model for top and bottom and then reduce. = = making it the end behavior model

  25. Limits approaching infinity analytically = = = - = 0

  26. Example of Squeeze Theorem

  27. HW • Pg. 54 # 19, 21, 23 (don’t sketch) • Pg. 67 # 5, 9, 17, 23, 27, 35, 37, 41, 43, 45, 51, 65, 95 – 98 • pg. 79 # 1, 3, 5, 17 • Pg. 88 # 1, 3 • Pg. 205 # 3 – 8, 15, 17, 21, 24, 25, 31, 34 • (you will need to use identity that tan x = )

  28. Continuity

  29. Investigating Continuity • Continuous Function • A function with a connected graph.

  30. Definition of Continuity • A function f is continuous at c if the following 3 conditions are met • 1) f(c) is defined • 2) • 3)

  31. Continuity • function is continuous on an interval IFF it is continuous at every point of the interval. • A continuous function is one that is continuous at every point IN ITS DOMAIN. • If a function is not continuous at a point, that is known as a point of discontinuity.

  32. Types of Discontinuities NON REMOVABLE • Jump Discontinuity • The one-sided limit exists, but have different values

  33. Types of Discontinuities NON REMOVABLE • Infinite Discontinuity • Let's say you have a function like f(x) = 1/x. • Then, as x goes to 0 from the right (x > 0), the function goes toward positive infinity. • As x goes to zero from the left (x < 0), the function goes toward negative infinity. • At x = 0, the function has no defined value. • We say that x = 0 is an infinite discontinuity, because the limits around the undefined point are infinite.

  34. Types of Discontinuities NON REMOVABLE • Oscillating Discontinuity • An oscillating discontinuity occurs at a value of x near to which a function refuses to settle down. • The function y = sin (1/x) has a discontinuity at x = 0 because it is not defined at x = 0.

  35. Types of Discontinuities • Removable discontinuity • A point on the graph that is undefined, or does not fit in with the rest of the graph. A “hole” in the graph. • The right-handed limit and the left-handed limit must be the same around the point of discontinuity. • A discontinuity is removable if f can be made continuous by appropriately defining (or redefining) f(c).

  36. Continuity on an open interval • A function is continuous on an open interval (a,b) if it is continuous at each point in the interval.

  37. Continuity on a closed interval • A function f is continuous on a closed interval [a,b] if it is continuous on the open interval (a,b) and • and • The function f is continuous from the right at a and continuous from the left at b

  38. Continuity of a composite function • If g is continuous at c and f is continuous at g(c), then the composite function given by (fog)(x) = f(g(x)) is continuous at c. • Example: f(x) = sinx and g(x) = • Discuss the continuity of the composite function h(x) = f(g(x)) • h(x) = sin . • This function is continuous for all values of x.

  39. Intermediate Value Theorem • If f is continuous on the closed interval [a,b] and w is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = w.

  40. Determining Continuity Algebraically • A function has a discontinuity at any point that makes the function undefined or indeterminate form on its domain.

  41. Example 1 • Use the graph to determine the limit, and discuss the continuity of the function. = 1 = 2 = DNE The function g(x) is not continuous because does not exist.

  42. Example 2 • Use the graph to determine the limit, and discuss the continuity of the function. = 2 f(1) = 3 = 2 = 2 The function g(x) is not continuous because

  43. Example 3 • Discuss the continuity of the function The function is continuous at every value of x except x = 2.

  44. Example 4 f(x) = The values that make the bottom zero are -2 and 5. = F(x) is continuous for all x values except x = -2 and 5. x = -2 is a removable discontinuity. f(x) = The values that make the bottom zero are 1 and -1. There is no way to factor and cancel anything. F(x) is continuous for all x values except x = 1 and -1. • Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable?

  45. Example 5 • Discuss the continuity of the composite function. h(x) = f(g(x)) • f(x) = g(x) = x – 1 • h(x) = • h(x) is continuous for all x > 1.

  46. Example 6 • Decide if the function is continuous at x = 1 • f(x) = { x³ + 1, x < 1 { x + 1, x ≥ 1 F(1) = Since f(1) the function is continuous 2 2 2 2

  47. HW • Pg. 79 # 25, 33, 35, 37, 41, 45, 47, 61, 63, 83, 85, 87, 89

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