Lesson 4: Limits Involving Infinity

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We examine two ways of extending the definition of limit: A function can be said to have a limit of infinity (or minus infinity) at a point if it grows without bound near that point.
A function can have a limit at a point if values of the function get close to a value as the points get arbitrarily large.

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Lesson 4: Limits Involving Infinity

  1. 1. Section 2.5 Limits Involving Infinity Math 1a February 4, 2008 Announcements Syllabus available on course website All HW on website now No class Monday 2/18 ALEKS due Wednesday 2/20
  2. 2. Outline Infinite Limits Vertical Asymptotes Infinite Limits we Know Limit “Laws” with Infinite Limits Indeterminate Limits Limits at Infinity Algebraic rates of growth Exponential rates of growth Rationalizing to get a limit Worksheet
  3. 3. Infinite Limits Definition The notation lim f (x) = ∞ x→a means that the values of f (x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a but not equal to a. Definition The notation lim f (x) = −∞ x→a means that the values of f (x) can be made arbitrarily large negative by taking x sufficiently close to a but not equal to a. Of course we have definitions for left- and right-hand infinite limits.
  4. 4. Vertical Asymptotes Definition The line x = a is called a vertical asymptote of the curve y = f (x) if at least one of the following is true: lim f (x) = ∞ lim f (x) = −∞ x→a x→a lim f (x) = ∞ lim f (x) = −∞ x→a+ x→a+ lim f (x) = ∞ lim f (x) = −∞ x→a− x→a−
  5. 5. Infinite Limits we Know 1 lim =∞ x→0+ x 1 lim = −∞ x→0− x 1 lim 2 = ∞ x→0 x
  6. 6. Finding limits at trouble spots Example Let t2 + 2 f (t) = t 2 − 3t + 2 Find lim f (t) and lim+ f (t) for each a at which f is not t→a− t→a continuous.
  7. 7. Finding limits at trouble spots Example Let t2 + 2 f (t) = t 2 − 3t + 2 Find lim f (t) and lim+ f (t) for each a at which f is not t→a− t→a continuous. Solution The denominator factors as (t − 1)(t − 2). We can record the signs of the factors on the number line.
  8. 8. − 0 + (t − 1) 1
  9. 9. − 0 + (t − 1) 1 − 0 + (t − 2) 2
  10. 10. − 0 + (t − 1) 1 − 0 + (t − 2) 2 + (t 2 + 2)
  11. 11. − 0 + (t − 1) 1 − 0 + (t − 2) 2 + (t 2 + 2) f (t) 1 2
  12. 12. − 0 + (t − 1) 1 − 0 + (t − 2) 2 + (t 2 + 2) + f (t) 1 2
  13. 13. − 0 + (t − 1) 1 − 0 + (t − 2) 2 + (t 2 + 2) + ±∞ f (t) 1 2
  14. 14. − 0 + (t − 1) 1 − 0 + (t − 2) 2 + (t 2 + 2) + ±∞ − f (t) 1 2
  15. 15. − 0 + (t − 1) 1 − 0 + (t − 2) 2 + (t 2 + 2) + ±∞ − ∞ f (t) 1 2
  16. 16. − 0 + (t − 1) 1 − 0 + (t − 2) 2 + (t 2 + 2) + ±∞ − ∞ + f (t) 1 2
  17. 17. Limit Laws with infinite limits To aid your intuition The sum of positive infinite limits is ∞. That is ∞+∞=∞ The sum of negative infinite limits is −∞. −∞ − ∞ = −∞ The sum of a finite limit and an infinite limit is infinite. a+∞=∞ a − ∞ = −∞
  18. 18. Rules of Thumb with infinite limits Don’t try this at home! The sum of positive infinite limits is ∞. That is ∞+∞=∞ The sum of negative infinite limits is −∞. −∞ − ∞ = −∞ The sum of a finite limit and an infinite limit is infinite. a+∞=∞ a − ∞ = −∞
  19. 19. Rules of Thumb with infinite limits The product of a finite limit and an infinite limit is infinite if the finite limit is not 0. ∞ if a > 0 a·∞= −∞ if a < 0. −∞ if a > 0 a · (−∞) = ∞ if a < 0. The product of two infinite limits is infinite. ∞·∞=∞ ∞ · (−∞) = −∞ (−∞) · (−∞) = ∞ The quotient of a finite limit by an infinite limit is zero: a = 0. ∞
  20. 20. Indeterminate Limits Limits of the form 0 · ∞ and ∞ − ∞ are indeterminate. There is no rule for evaluating such a form; the limit must be examined more closely.
  21. 21. Indeterminate Limits Limits of the form 0 · ∞ and ∞ − ∞ are indeterminate. There is no rule for evaluating such a form; the limit must be examined more closely. 1 Limits of the form are also indeterminate. 0
  22. 22. Outline Infinite Limits Vertical Asymptotes Infinite Limits we Know Limit “Laws” with Infinite Limits Indeterminate Limits Limits at Infinity Algebraic rates of growth Exponential rates of growth Rationalizing to get a limit Worksheet
  23. 23. Definition Let f be a function defined on some interval (a, ∞). Then lim f (x) = L x→∞ means that the values of f (x) can be made as close to L as we like, by taking x sufficiently large.
  24. 24. Definition Let f be a function defined on some interval (a, ∞). Then lim f (x) = L x→∞ means that the values of f (x) can be made as close to L as we like, by taking x sufficiently large. Definition The line y = L is a called a horizontal asymptote of the curve y = f (x) if either lim f (x) = L or lim f (x) = L. x→∞ x→−∞
  25. 25. Definition Let f be a function defined on some interval (a, ∞). Then lim f (x) = L x→∞ means that the values of f (x) can be made as close to L as we like, by taking x sufficiently large. Definition The line y = L is a called a horizontal asymptote of the curve y = f (x) if either lim f (x) = L or lim f (x) = L. x→∞ x→−∞ y = L is a horizontal line!
  26. 26. Theorem Let n be a positive integer. Then 1 lim n = 0 x→∞ x 1 lim =0 x→−∞ x n
  27. 27. Using the limit laws to compute limits at ∞ Example Find 2x 3 + 3x + 1 lim x→∞ 4x 3 + 5x 2 + 7 if it exists. A does not exist B 1/2 C 0 D ∞
  28. 28. Using the limit laws to compute limits at ∞ Example Find 2x 3 + 3x + 1 lim x→∞ 4x 3 + 5x 2 + 7 if it exists. A does not exist B 1/2 C 0 D ∞
  29. 29. Solution Factor out the largest power of x from the numerator and denominator. We have 2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 ) = 3 4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 ) 2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3 lim = lim x→∞ 4x 3 + 5x 2 + 7 x→∞ 4 + 5/x + 7/x 3 2+0+0 1 = = 4+0+0 2
  30. 30. Solution Factor out the largest power of x from the numerator and denominator. We have 2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 ) = 3 4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 ) 2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3 lim = lim x→∞ 4x 3 + 5x 2 + 7 x→∞ 4 + 5/x + 7/x 3 2+0+0 1 = = 4+0+0 2 Upshot When finding limits of algebraic expressions at infinitely, look at the highest degree terms.
  31. 31. Another Example Example Find √ 3x 4 + 7 lim x→∞ x2 + 3
  32. 32. Another Example Example Find √ 3x 4 + 7 lim x→∞ x2 + 3 Solution √ The limit is 3.
  33. 33. Example x2 Make a conjecture about lim . x→∞ 2x
  34. 34. Example x2 Make a conjecture about lim . x→∞ 2x Solution The limit is zero. Exponential growth is infinitely faster than geometric growth
  35. 35. Rationalizing to get a limit Example Compute lim 4x 2 + 17 − 2x . x→∞
  36. 36. Rationalizing to get a limit Example Compute lim 4x 2 + 17 − 2x . x→∞ Solution This limit is of the form ∞ − ∞, which we cannot use. So we rationalize the numerator (the denominator is 1) to get an expression that we can use the limit laws on.
  37. 37. Outline Infinite Limits Vertical Asymptotes Infinite Limits we Know Limit “Laws” with Infinite Limits Indeterminate Limits Limits at Infinity Algebraic rates of growth Exponential rates of growth Rationalizing to get a limit Worksheet
  38. 38. Worksheet

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