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# 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