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limits and continuity

- 1. C O N T I N U I T Y TOPIC 8 C A L C U L U S I W I T H A N A L Y T I C G E O M E T R Y
- 2. S U B T O P I C S ❑ Definition of Continuity ❑ Types of Continuity and Discontinuity ❑ Continuity of a Composite Function ❑ Continuity Interval
- 3. Overview Many functions have the property that their graphs can be traced with a pencil without lifting the pencil from the page. Such functions are called continuous. The property of continuity Is exhibited by various aspects of nature. The water flow in the river is continuous. The flow of time in human life is continuous, you are getting older continuously. And so on. Similarly, in Mathematics, we have the notion of continuity of a function. What it simply mean is that a functions is said to be continuous of you can sketch its curve on a graph without lifting your pen even once. It is a very straight forward and close to accurate definition actually. But for the sake of higher mathematics, we must define it in a more precise
- 4. Overview ● The limit of a function as x approaches a can often be found simply by calculating the value of the function at a. Functions with this property are called continuous at a. ● We will see that the mathematical definition of continuity corresponds closely with the meaning of the word continuity in everyday language. (A continuous process is one that takes place gradually, without interruption or abrupt change.)
- 5. Overview ● In fact, the change in f (x) can be kept as small as we please by keeping the change in x sufficiently small. ● If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not continuous at a. ● Physical phenomena are usually continuous. For instance, the displacement or velocity of a vehicle varies continuously with time, as does a person’s height. But discontinuities do occur in such situations as electric currents.
- 6. Overview ● In fact, the change in f (x) can be kept as small as we please by keeping the change in x sufficiently small. ● If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not continuous at a. ● Physical phenomena are usually continuous. For instance, the displacement or velocity of a vehicle varies continuously with time, as does a person’s height. But discontinuities do occur in such situations as electric currents.
- 7. Overview ● Geometrically, you can think of a function that is continuous at every number in an interval as a function whose graph has no break in it. The graph can be drawn without removing your pen from the paper.
- 8. DEFINITION OF CONTINUITY A function f is continuous at x if it satisfies the following condition: Definition of Continuity A function f is continuous at x=a when: 1. 𝟏. 𝒇 𝒂 𝐢𝐬 𝐝𝐞𝐟𝐢𝐧𝐞𝐝 2. 2. 𝐥𝐢𝐦 𝒙→𝒂 𝒇 𝒙 𝒆𝒙𝒊𝒔𝒕 3. 3. 𝐥𝐢𝐦 𝒙→𝒂 𝒇 𝒙 = 𝒇(𝒂) If any one of the condition is not met, the function in not continuous at x=a The definition says that f is continuous at a if f (x) approaches f (a) as x approaches a. Thus a continuous function f has the property that a small change in x produces only a small change in f (x).
- 9. Illustrative Examples lim 𝑥→1 𝑓(𝑥) = 𝑥2 + 𝑥 + 1 Checking of Continuity Is the function defined at x = 1? Yes Does the limit of the function as x approaches 1 exist? Yes Does the limit of the function as x approaches 0 equal the function value at x = 1? Yes
- 10. Illustrative Examples lim 𝑥→1 𝑓(𝑥) = 𝑥2 + 𝑥 + 1 Checking of Continuity Is the function defined at x = 1? Yes Does the limit of the function as x approaches 1 exist? Yes Does the limit of the function as x approaches 1 equal the function value at x = 1? Yes x 0.9 0.99 0.999 1 1.01 1.001 1.0001 f(x) 2.71 2.97 2.99 3 3.03 3.003 3.0012 ∴ The function is continuous at 1.
- 11. Illustrative Examples lim 𝑥→1 𝑥2 − 2𝑥 + 3 Checking of Continuity Is the function defined at x = 1? Yes Does the limit of the function as x approaches 1 exist? Yes Does the limit of the function as x approaches 0 equal the function value at x = 1? Yes
- 12. Illustrative Examples Checking of Continuity Is the function defined at x = 1? Yes Does the limit of the function as x approache s 1 exist? Yes Does the limit of the function as x approache s 1 equal the function value at x = 1? Yes x 0.9 0.99 0.999 1 1.01 1.001 1.0001 f(x) 2.01 2.0001 2.000001 2 2.0001 2.000001 2.00000001 ∴ The function is continuous at 1. lim 𝑥→1 𝑥2 − 2𝑥 + 3
- 13. Illustrative Examples lim 𝑥→1 𝑥3 − 𝑥 Checking of Continuity Is the function defined at x = 1? Yes Does the limit of the function as x approaches 1 exist? Yes Does the limit of the function as x approaches 0 equal the function value at x = 1? Yes
- 14. Illustrative Examples Checking of Continuity Is the function defined at x = 1? Yes Does the limit of the function as x approache s 1 exist? Yes Does the limit of the function as x approache s 1 equal the function value at x = 1? Yes x 1.9 1.99 1.999 2 2.01 2.001 2.0001 f(x) 4.96 5.89 5.98 6 6.11 6.011 6.0011 ∴ The function is continuous at 2. lim 𝑥→2 𝑥3 − 𝑥
- 15. DISCONTINUITY TOPIC 9 C A L C U L U S I W I T H A N A L Y T I C G E O M E T R Y
- 16. D I S C O N T I N U I T Y Discontinuity: a point at which a function is not continuous
- 17. D I S C O N T I N U I T Y Discontinuity: a point at which a function is not continuous
- 18. D I S C O N T I N U I T Y JUMP DISCONTINUITY Jump Discontinuities: both one-sided limits exist, but have different values. The graph of f(x) below shows a function that is discontinuous at x=a. In this graph, you can easily see that f(x)=L f(x)=M The function is approaching different values depending on the direction x is coming from. When this happens, we say the function has a jump discontinuity at x=a.
- 19. D I S C O N T I N U I T Y ● Functions with jump discontinuities, when written out mathematically, are called piecewise functions because they are defined piece by piece. ● Piecewise functions are defined on a sequence of intervals. ● You'll see how the open and closed circles come into play with these functions. Let's look at a function now to see what a piecewise function looks like.
- 20. Illustrative Examples 𝑓 𝑥 = ቊ 𝑥 + 1, 𝑥 < −1 𝑥 + 4, 𝑥 ≥ −1 Left Side Limit 𝒂− 𝑥 + 1, 𝑥 < −1 Right Side Limit 𝒂+ 𝑥 + 4, 𝑥 ≥ −1 −4 −3 −2 −1 0 1 −3 −2 −1 3 4 5
- 21. Illustrative Examples 𝑓 𝑥 = ቊ 𝑥 + 1, 𝑥 < −1 𝑥 + 4, 𝑥 ≥ −1 Left Side Limit 𝒂− 𝑥 + 1, 𝑥 < −1 Right Side Limit 𝒂+ 𝑥 + 4, 𝑥 ≥ −1 x −4 −3 −2 −1 0 1 f(x) −3 −2 −1 3 4 5
- 22. D I S C O N T I N U I T Y INFINITE DISCONTINUITY The graph on the right shows a function that is discontinuous at x=a. The arrows on the function indicate it will grow infinitely large as x approaches a. Since the function doesn't approach a particular finite value, the limit does not exist. This is an infinite discontinuity. The following two graphs are also examples of infinite discontinuities at x=a. Notice that in all three cases, both of the one-sided limits are infinite.
- 23. D I S C O N T I N U I T Y REMOVABLE DISCONTINUITY Two Types of Discontinuities 1) Removable (hole in the graph) 2) Non-removable (break or vertical asymptote) A discontinuity is called removable if a function can be made continuous by defining (or redefining) a point.
- 24. D I S C O N T I N U I T Y DIFFERENCE BETWEEN A REMOVABLE AND NON REMOVABLE DISCONTINUITY If the limit does not exist, then the discontinuity is non– removable. In essence, if adjusting the function’s value solely at the point of discontinuity will render the function continuous, then the discontinuity is removable.
- 25. D I S C O N T I N U I T Y REMOVABLE DISCONTINUITY Step 1: Factor the numerator and the denominator. Step 2: Identify factors that occur in both the numerator and the denominator. Step 3: Set the common factors equal to zero. Step 4: Solve for x.
- 26. Illustrative Examples lim 𝑥→3 𝑓 𝑥 = 𝑥2 − 9 𝑥 − 3 Step 1: Factor the numerator and the denominator. Step 2: Identify factors that occur in both the numerator and the denominator. Step 3: Set the common factors equal to zero. Step 4: Solve for x. (𝑥 + 3)(𝑥 − 3) 𝑥 − 3 𝑥 + 3 𝑓 𝑥 = 3 + 3 lim 𝑥→3 𝑓 𝑥 = 𝑥2 − 9 𝑥 − 3 = 6
- 27. Illustrative Examples lim 𝑥→3 𝑓 𝑥 = 𝑥2 − 9 𝑥 − 3 Left Side Limit 𝒂− 𝑥 < 3 Right Side Limit 𝒂+ 𝑥 > 3 2.9 2.99 2.999 3.01 3.001 3.0001 −5.9 −5.99 −5.999 6.01 6.001 6.0001
- 28. Illustrative Examples lim 𝑥→3 𝑓 𝑥 = 𝑥2 − 9 𝑥 − 3 Left Side Limit 𝒂− 𝑥 < 3 Right Side Limit 𝒂+ 𝑥 > 3 2.9 2.99 2.999 3.01 3.0013.0001 5.9 5.99 5.999 6.01 6.001 6.0 001
- 29. Illustrative Examples lim 𝑥→2 𝑓 𝑥 = 𝑥2 − 5𝑥 + 4 𝑥2 − 4 Step 1: Factor the numerator and the denominator. Step 2: Identify factors that occur in both the numerator and the denominator. Step 3: Set the common factors equal to zero. Step 4: Solve for x. (𝑥 + 5)(𝑥 − 1) (𝑥 + 2)(𝑥 − 2) Non-removable, the limits DNE.
- 30. Continuity of a Composite Function THEOREM CONTINUITY OF A COMPOSITE FUNCTION If g is a continuous at a and f is a continuous at g(a) ,then the composite function 𝑓°𝑔 𝑥 = 𝑓(𝑔 𝑥 ) is continuous at a. Proof: Because g is continuous at a, 𝑔(𝑥) = 𝑔(𝑎) or, equivalently Now f is continuous at g(a); thus, we apply Theorem to the composite 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝑓°𝑔 𝑥 = 𝑓( 𝑔(𝑥)) = 𝑓(𝑔(𝑎) = (𝑓°𝑔)(𝑎)
- 31. Illustrative Example: Let f 𝑥 = 𝑥2 𝑎𝑛𝑑 𝑔 𝑥 = 𝑥 + 1 𝑓 𝑔 𝑥 = (𝑥 + 1)2 lim 𝑥→2 (𝑥 + 1)2 = (2 + 1)2 = 9 lim 𝑥→2 (𝑥 + 1)2 = 9 Illustrative Example: Discuss the continuity of 𝑓(𝑔 𝑥 𝑤ℎ𝑒𝑟𝑒 f 𝑥 = 1 𝑥 𝑎𝑛𝑑 𝑔 𝑥 = 𝑥2 − 1 𝑓 𝑔 𝑥 = (𝑥 + 1)2 lim 𝑥→2 (𝑥 + 1)2 = (2 + 1)2 = 9 lim 𝑥→2 (𝑥 + 1)2 = 9
- 32. C O N T I N U I T Y OF AN INTERVAL TOPIC 10 C A L C U L U S I W I T H A N A L Y T I C G E O M E T R Y
- 33. CONTINUITY OF AN INTERVAL A function is continuous on an interval if and only if we can trace the graph of the function without lifting our pen on the given interval
- 34. CONTINUITY OF AN INTERVAL An interval is a set of real numbers between two given numbers called the endpoints of the interval Finite Interval- intervals whose endpoints are bounded An open interval is one that does not include its endpoints: 𝑎, 𝑏 𝑜𝑟 𝑎 < 𝑥 < 𝑏 A closed interval is one that includes its endpoints 𝑎, 𝑏 𝑜𝑟 𝑎 ≤ 𝑥 ≤ 𝑏 Combination: composed of an open and closed interval on either side: ሾ𝑎, 𝑏) 𝑜𝑟 𝑎 ≤ 𝑥 < 𝑏; ( ሿ 𝑎, 𝑏 𝑜𝑟 𝑎 < 𝑥 ≤ 𝑏
- 35. CONTINUITY OF AN INTERVAL Infinite Interval These are intervals with at least one unbounded side. Open Left-bounded: 𝑎, ∞ 𝑜𝑟 𝑥 > 𝑎 Left side has an endpoint up to positive infinity Close Left bounded: ሾ𝑎, ∞) 𝑜𝑟 𝑥 ≥ 𝑎 Open Right Bounded: −∞, 𝑏 𝑜𝑟 𝑥 < 𝑏 Close Right Bounded: ( ሿ −∞, 𝑏 𝑜𝑟 𝑥 ≤ 𝑏 Unbounded: −∞, ∞ 𝑜𝑟 𝑠𝑒𝑡 𝑜𝑓 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
- 36. Interval Notation Set Notation (2, 5) 2 < 𝑥 < 5 2, 5 2 ≤ 𝑥 ≤ 5 ( ሿ 2,5 2 < 𝑥 ≤ 5 ሾ2,5) 2 ≤ 𝑥 < 5 −∞, 2 x<2 ( ሿ −∞, 2 x≤ 2 (5, ∞) x>5 ሾ5, ∞) 𝑥 ≥ 5 −∞, ∞ All Real Number
- 37. Continuity on an Open Interval A function is continuous on an open interval if it is continuous for any real number on that interval Continuity on an Closed Interval A function f is continuous on a closed interval 𝑎, 𝑏 if it is continuous on (a, b) and lim 𝑥→𝑎= 𝑓 𝑥 = 𝑓 𝑎 𝑎𝑛𝑑 lim 𝑥→𝑏− 𝑓 𝑥 = 𝑓(𝑏)
- 38. Determine if the function is continuous or not 1. 𝑓 𝑥 = 3𝑥 − 6, ( ሿ −∞, 1 𝑥 ≤ 1 𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
- 39. Determine if the function is continuous or not 2. ℎ 𝑥 𝑥−5 𝑥2−𝑥−6 −1,4 −1 < 𝑥 < 4 Solve for x. 𝑥2 − 𝑥 − 6 = 0, 𝑥 − 3 𝑥 + 2 = 0 𝑥 = 3, 𝑥 = −2, x ≠ 3, 𝑥 ≠ −2 Discontinuous on the given interval
- 40. Determine if 𝑓 𝑥 = ቊ 2𝑥 − 1, 𝑥 < 5 𝑥2 , 𝑥 ≥ 5 is continuous on 2,5 𝑤ℎ𝑒𝑟𝑒 𝑎 = 2, 𝑏 = 5 Step 1: Check if continuous on (2,5) We’ll us the subfunction that satisfy the interval Use f x = 2𝑥 − 1 Continuous for all real numbers
- 41. Continuity on an Closed Interval A function f is continuous on a closed interval 𝑎, 𝑏 if it is continuous on (a, b) and lim 𝑥→𝑎+= 𝑓 𝑥 = 𝑓 𝑎 𝑎𝑛𝑑 lim 𝑥→𝑏− 𝑓 𝑥 = 𝑓(𝑏)
- 42. Determine if 𝑓 𝑥 = ቊ 2𝑥 − 1, 𝑥 < 5 𝑥2 , 𝑥 ≥ 5 is continuous on 2,5 𝑤ℎ𝑒𝑟𝑒 𝑎 = 2, 𝑏 = 5 Step 2: Check if lim 𝑥→𝑎+ 𝑓 𝑥 = 𝑓(𝑎) lim 𝑥→2+ 𝑓 𝑥 = 2𝑥 − 1 = 2 2 − 1 = 3 Check 𝑓 2 = 2𝑥 − 1 = 2 2 − 1 = 3
- 43. Determine if 𝑓 𝑥 = ቊ 2𝑥 − 1, 𝑥 < 5 𝑥2 , 𝑥 ≥ 5 is continuous on 2,5 𝑤ℎ𝑒𝑟𝑒 𝑎 = 2, 𝑏 = 5 Step 3: Check if lim 𝑥→𝑏− 𝑓 𝑥 = 𝑓(𝑏) lim 𝑥→5− 𝑓 𝑥 = 2𝑥 − 1 = 2 5 − 1 = 9 Check 𝑓 5 = 𝑥2 = 52 = 25 Discontinuous on the interval 2,5
- 44. Determine if 𝑓 𝑥 = ൝ 𝑥 − 6, 𝑥 < 2 𝑥+1 𝑥−1 𝑥 ≥ 2 is continuous on −3,1 𝑤ℎ𝑒𝑟𝑒 𝑎 = −3 𝑏 = 1 Step 1: Check if continuous on (-3,1) We’ll us the subfunction that satisfy the interval Use f x = 𝑥 − 6 Continuous for all real numbers
- 45. Determine if 𝑓 𝑥 = ൝ 𝑥 − 6, 𝑥 < 2 𝑥+1 𝑥−1 𝑥 ≥ 2 is continuous on −3,1 𝑤ℎ𝑒𝑟𝑒 𝑎 = −3 𝑏 = 1 Step 2: Check if lim 𝑥→𝑎+ 𝑓 𝑥 = 𝑓(𝑎) lim 𝑥→−3+ 𝑓 𝑥 = 𝑥 − 6 = −3 − 6 = 9 Check 𝑓 −3 = 𝑥 − 6 = −3 − 6 = 9
- 46. Step 3: Check if lim 𝑥→𝑏− 𝑓 𝑥 = 𝑓(𝑏) lim 𝑥→1− 𝑓 𝑥 = 𝑥 − 6 = 1 − 6 = −5 Check 𝑓 1 = 𝑥 − 6 = 1 − 6 = −5 Continuous on the interval −3,1 Determine if 𝑓 𝑥 = ൝ 𝑥 − 6, 𝑥 < 2 𝑥+1 𝑥−1 𝑥 ≥ 2 is continuous on −3,1 𝑤ℎ𝑒𝑟𝑒 𝑎 = −3 𝑏 = 1
- 47. Determine if 𝑓 𝑥 = ቊ𝑥2 − 9, 𝑥 ≥ 4 𝑥 + 5 < 4 is continuous on 1,4 𝑤ℎ𝑒𝑟𝑒 𝑎 = 1 𝑏 = 4 Step 1: Check if continuous on (1,4) We’ll us the subfunction that satisfy the interval Use f x = 𝑥 + 5 Continuous for all real numbers
- 48. Determine if 𝑓 𝑥 = ቊ𝑥2 − 9, 𝑥 ≥ 4 𝑥 + 5 < 4 is continuous on 1,4 𝑤ℎ𝑒𝑟𝑒 𝑎 = 1 𝑏 = 4 Step 2: Check if lim 𝑥→𝑎+ 𝑓 𝑥 = 𝑓(𝑎) lim 𝑥→1+ 𝑓 𝑥 = 𝑥 + 5 = 1 + 5 = 6 Check 𝑓 1 = 𝑥 + 5 = 1 + 5 = 6
- 49. Determine if 𝑓 𝑥 = ቊ𝑥2 − 9, 𝑥 ≥ 4 𝑥 + 5 < 4 is continuous on 1,4 𝑤ℎ𝑒𝑟𝑒 𝑎 = 1 𝑏 = 4 Step 3: Check if lim 𝑥→𝑏− 𝑓 𝑥 = 𝑓(𝑏) lim 𝑥→4− 𝑓 𝑥 = 𝑥 + 5 = 4 + 9 = 9 Check 𝑓 4 = 𝑥2 − 9 = 42 − 9 = 16 − 9 = 7 Discontinuous on the interval 1,4