1. AP CALCULUS 1003 Limits pt.3 Limits at Infinity and End Behavior

2. REVIEW: ALGEBRA is a ________________________ machine that ___________________ a function ___________ a point. CALCULUS is a ________________________ machine that ___________________________ a function ___________ a point Function Evaluates at Limit Describes the behavior of near

3. END BEHAVIOR

4. LIMITS AT INFINITY Part 2: End Behavior GENERAL IDEA: The behavior of a function as x gets very large ( in a positive or negative direction)

5. END Behavior: Limit Layman’s Description: Notation: Horizontal Asymptotes: Note: m Closer to L than ε If it has a limit = L then the HA y=L

6. GNAW: Graphing EX: If you cover the middle what happens? 2 -2 3 3

7. GNAW: Algebraic Method: DIRECT SUBSTITUTION gives a second INDETERMINANT FORM Theorem: Method:Divide by largest degree in denominator

8. End Behavior Models EX: (with Theorem) = End behavior HA y=2 0 0 0 0 0 0 0 0

9. End Behavior Models Summary: ________________________________________ A). B). C). Leading term test (reduce leading term) If degree on bottom is largest limit = 0 If the degrees are the same then the limit = reduced fraction If the degrees on top is larger limit DNE but EB acts like reduced power function E.B. HA y=0

10. Continuity

11. General Idea: General Idea: ________________________________________ We already know the continuity of many functions: Polynomial (Power), Rational, Radical, Exponential, Trigonometric, and Logarithmic functions DEFN: A function is continuous on an interval if it is continuous at each point in the interval. DEFN: A function is continuous at a point IFF a) b) c) Can you draw without picking up your pencil Has a point f(a) exists Has a limit Limit = value

12. Continuity Theorems

13. Continuity on a CLOSED INTERVAL. Theorem: A function is Continuous on a closed interval if it is continuous at every point in the open interval and continuous from one side at the end points. Example : The graph over the closed interval [-2,4] is given. From the right From the left

14. Discontinuity No value f(a) DNE hole jump Limit does not equal value Limit ≠ value Vertical asymptote a) c) b)

15. Discontinuity: cont. Method: (a). (b). (c). Removable or Essential Discontinuities Test the value =Look for f(a) = Test the limit Holes and hiccups are removable Jumps and Vertical Asymptotes are essential Test f(a) =f(a) = Lim DNE Jump = cont. ≠ hiccup

16. Examples: EX: Identify the x-values (if any) at which f(x)is not continuous. Identify the reason for the discontinuity and the type of discontinuity. Is the discontinuity removable or essential? removable or essential? = x≠ 4 Hole discontinuous because f(x) has no value It is removable

17. Examples: cont. Identify the x-values (if any) at which f(x)is not continuous. Identify the reason for the discontinuity and the type of discontinuity. Is the discontinuity removable or essential? removable or essential? x≠3 VA discontinuous because no value It is essential

18. Examples: cont. Identify the x-values (if any) at which f(x)is not continuous. Identify the reason for the discontinuity and the type of discontinuity. Is the discontinuity removable or essential? Step 1: Value must look at 4 equation f(1) = 4 Step 2: Limit It is a jump discontinuity(essential) because limit does not exist

19. Graph: Determine the continuity at each point. Give the reason and the type of discontinuity. x = -3 x = -2 x = 0 x =1 x = 2 x = 3 Hole discont. No value VA discont. Because no value no limit Hiccup discont. Because limit ≠ value Continuous limit = value VA discont. No limit Jump discont. Because limit DNE

20. Algebraic Method a. b. c. Value:f(2) = 8 Look at function with equal Limit: Limit = value: 8=8

21. Algebraic Method At x=1 a. b. c. At x=3 a. b. Limit c. Value: Limit: Jump discontinuity because limit DNE, essential Value: Hole discontinuity because no value, removable No further test necessary f(1) = -1

22. Rules for Finding Horizontal Asymptotes 1. If degree of numerator < degree of denominator, horizontal asymptote is the line y=0 (x axis) 2. If degree of numerator = degree of denominator, horizontal asymptote is the line y = ratio of leading coefficients. 3. If degree of numerator > degree of denominator, there is no horizontal asymptote, but possibly has an oblique or slant asymptote.

23. Consequences of Continuity: A. INTERMEDIATE VALUE THEOREM ** Existence Theorem EX: Verify the I.V.T. for f(c) Then find c. on If f© is between f(a) and f(b) there exists a c between a and b c ab f(a) f(c) f(b) f(1) =1 f(2) = 4 Since 3 is between 1 and 4. There exists a c between 1 and 2 such that f(c) =3 x 2 =3 x=±1.732

24. Consequences: cont. EX: Show that the function has a ZERO on the interval [0,1]. I.V.T - Zero Locator Corollary CALCULUS AND THE CALCULATOR: The calculator looks for a SIGN CHANGE between Left Bound and Right Bound f(0) = -1 f(1) = 2 Since 0 is between -1 and 2 there exists a c between 0 and 1 such that f(c) = c Intermediate Value Theorem

25. Consequences: cont. EX: I.V.T - Sign on an Interval - Corollary (Number Line Analysis) 0 1 2 3 4 5 -2 -3-4 -5 We know where the zeroes are located Choose a point between them to determine whether the graph is positive or negative

26. Consequences of Continuity: B. EXTREME VALUE THEOREM On every closed interval there exists an absolute maximum value and minimum value.