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UNIT 1B LESSON 8

UNIT 1B LESSON 8. Limits of Infinite Sequences. Important stuff coming! . Limits of Infinite Sequences. A sequence that does not have a last term is called infinite . . Look at the following sequence:  ½ , ( ½ ) 2 , ( ½ ) 3 , ( ½ ) 4 . . . (½) x . . . or

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UNIT 1B LESSON 8

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  1. UNIT 1B LESSON 8 Limits of Infinite Sequences

  2. Important stuff coming!  Limits of Infinite Sequences A sequence that does not have a last term is called infinite.  Look at the following sequence:  ½ , ( ½ )2 , ( ½ )3 , ( ½ )4 . . . (½)x . . . or ½ , ¼ , ⅛ , 1/16 . . . (½)x. . .  How small do these terms get?  Use your calculator and put in some large values for x and see what happens.  Obviously, the numbers are getting smaller.  Is there a number that the sequence approaches? 

  3. (1, ½) (2, ¼ ) (3, 1/8 ) (4, 1/16) (5,1/32) The Graph of an Infinite Sequence Take a look at the graph of this sequence. You can see that the graph is asymptotically approaching the x-axis.  This means that the sequence approaches the number zero!!  You have just found your first limit as x approaches infinity. Conclusion:  lim ( ½ )x= 0 x → ∞

  4. Lesson #11 Worksheet Examples 1. State the limits of the following sequences, or state that the limit does not exist. 0 a) 4 b) ∞ c) 3 d) 0 e) ,… f) 6 g) does not exist

  5. (Note: infinity is not a number.  It means that the sequence Increases or decreases without bound!)  Vertical asymptote at x = 0 - 0.1 0.1 - 0.01 0.01 ∞ -∞ 0.001 - 0.001 - 0.0001 0.0001 - 0.00001 0.00001 Horizontal asymptote at y = 0

  6. Study these examples closely as it will lead to an important theorem!!  Use your calculator, and plug in increasing values for x.    Do you see that as you use higher and higher values for x, the sequence approaches zero!!  This leads us to an important theorem and a method to calculate limits!

  7. This theorem implies that any fraction having x to a positive power in the denominator (power in the denominator is greater than the power of x in the numerator) will approach zero!!

  8. Lesson #11 Worksheet Examples 2. Find the following limits , , a) … , , = 6 , , … , , , b) = 0 , , , ∞ 5, 10, 15, 20, 25, … c)

  9. d) , , , … , , … , = 0 e) , , 1 , , , , = 0 ∞ 2, 12, 36, 80 … f)

  10. THINK ABOUT THIS or divide every term by 4 first or divide every term by 22 first This gives us a way to calculate some limits.  If the problem is a fraction, try dividing the numerator and denominator by the highest power of x in the denominator!

  11. Lesson #8 Worksheet Examples 3a. Find the following limits

  12. Lesson #8 Worksheet Examples 3b. Find the following limits

  13. Limits to Infinity and Graphs of Rational Functions EXAMPLE Since division by 0 is undefined OR nor limit will exist at x = These restrictions will give the graph 2 vertical asymptotes at and

  14. EXAMPLE Find To find the limit, divide top and bottom by x2 Notice that both and approach zero because of the previous theorem!! The value of this limit will give the graph a horizontal asymptote at

  15. EXAMPLE Set your window at [-10, 10,1] [-3, 3, 1] and graph Vertical Asymptotes Horizontal Asymptote

  16. Lesson #8 Worksheet Examples • x = 5 4. • y = 1 • x = 5 Vertical asymptote at Horizontal asymptote at • y = 1

  17. Lesson #8 Worksheet Examples 5. • 1 Vertical asymptote at x = Vertical asymptote at Horizontal asymptote at y=

  18. Lesson #8 Worksheet Examples 6. Since x2 + 1 ≠ 0 there are no vertical asymptotes • 0 Horizontal asymptote at y =

  19. Lesson #8 Worksheet Examples 7. Vertical asymptote at Horizontal asymptote at

  20. Lesson #8 Worksheet Examples 8. Solid point at

  21. Lesson #8 Worksheet Examples 8. ‘Hole’ at

  22. Lesson #8 Worksheet Examples 8. DNE Vertical asymptote at

  23. Lesson #8 Worksheet Examples 8. • y = Horizontal asymptote at

  24. Asymptotes will be examined in detail in a later chapter on curve sketching

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