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- 1. Section 1.6 Limits involving Infinity V63.0121.021, Calculus I New York University September 23, 2010 Announcements Quiz 1 is next week in recitation. Covers Sections 1.1–1.4 . . . . . .
- 2. Announcements Quiz 1 is next week in recitation. Covers Sections 1.1–1.4 . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 2 / 38
- 3. Objectives “Intuit” limits involving infinity by eyeballing the expression. Show limits involving infinity by algebraic manipulation and conceptual argument. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 3 / 38
- 4. Recall the definition of limit Definition We write lim f(x) = L x→a and say “the limit of f(x), as x approaches a, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 4 / 38
- 5. Recall the unboundedness problem 1 Recall why lim+ doesn’t exist. x→0 x y . .?. L . x . No matter how thin we draw the strip to the right of x = 0, we cannot “capture” the graph inside the box. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 5 / 38
- 6. Recall the unboundedness problem 1 Recall why lim+ doesn’t exist. x→0 x y . .?. L . x . No matter how thin we draw the strip to the right of x = 0, we cannot “capture” the graph inside the box. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 5 / 38
- 7. Recall the unboundedness problem 1 Recall why lim+ doesn’t exist. x→0 x y . .?. L . x . No matter how thin we draw the strip to the right of x = 0, we cannot “capture” the graph inside the box. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 5 / 38
- 8. Recall the unboundedness problem 1 Recall why lim+ doesn’t exist. x→0 x y . .?. L . x . No matter how thin we draw the strip to the right of x = 0, we cannot “capture” the graph inside the box. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 5 / 38
- 9. Outline Infinite Limits Vertical Asymptotes Infinite Limits we Know Limit “Laws” with Infinite Limits Indeterminate Limit forms Limits at ∞ Algebraic rates of growth Rationalizing to get a limit . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 6 / 38
- 10. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that 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. . x . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
- 11. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that 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. . x . “Large” takes the place of “close to L”. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
- 12. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that 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. . x . “Large” takes the place of “close to L”. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
- 13. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that 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. . x . “Large” takes the place of “close to L”. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
- 14. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that 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. . x . “Large” takes the place of “close to L”. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
- 15. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that 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. . x . “Large” takes the place of “close to L”. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
- 16. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that 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. . x . “Large” takes the place of “close to L”. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
- 17. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that 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. . x . “Large” takes the place of “close to L”. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
- 18. Negative Infinity Definition The notation lim f(x) = −∞ x→a means that the values of f(x) can be made arbitrarily large negative (as large as we please) by taking x sufficiently close to a but not equal to a. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 8 / 38
- 19. Negative Infinity Definition The notation lim f(x) = −∞ x→a means that the values of f(x) can be made arbitrarily large negative (as large as we please) by taking x sufficiently close to a but not equal to a. We call a number large or small based on its absolute value. So −1, 000, 000 is a large (negative) number. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 8 / 38
- 20. 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− . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 9 / 38
- 21. Infinite Limits we Know y . . . 1 lim+ = ∞ x→0 x . . . . . . . . x . . . . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 10 / 38
- 22. Infinite Limits we Know y . . . 1 lim+ = ∞ x→0 x . 1 lim = −∞ x→0− x . . . . . . . x . . . . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 10 / 38
- 23. Infinite Limits we Know y . . . 1 lim+ = ∞ x→0 x . 1 lim = −∞ x→0− x . . . . . . . x . 1 lim =∞ x→0 x2 . . . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 10 / 38
- 24. Finding limits at trouble spots Example Let x2 + 2 f(x) = x2 − 3x + 2 Find lim f(x) and lim+ f(x) for each a at which f is not continuous. x→a− x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 11 / 38
- 25. Finding limits at trouble spots Example Let x2 + 2 f(x) = x2 − 3x + 2 Find lim f(x) and lim+ f(x) for each a at which f is not continuous. x→a− x→a Solution The denominator factors as (x − 1)(x − 2). We can record the signs of the factors on the number line. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 11 / 38
- 26. Use the number line . . x − 1) ( . . . . . . . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
- 27. Use the number line − .. 0 .. . + . x − 1) . . ( 1 . . . . . . . . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
- 28. Use the number line − .. 0 .. . + . x − 1) . . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . . . . . . . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
- 29. Use the number line − .. 0 .. . + . x − 1) . . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( . . . . . . . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
- 30. Use the number line − .. 0 .. . + . x − 1) . . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( . . .. . . .. . f .(x) 1 . 2 . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
- 31. Use the number line − .. 0 .. . + . x − 1) . . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( .+ . .. . . .. . f .(x) 1 . 2 . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
- 32. Use the number line . mall − .. 0 .. . + s . x − 1) . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( .+ +∞ . . . . . .. . f .(x) 1 . 2 . lim f(x) = + ∞ x→1− . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
- 33. Use the number line . mall − .. 0 .. . + s . x − 1) . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( .+ +∞ . . ∞ . . − . .. . f .(x) 1 . 2 . lim f(x) = + ∞ x→1− lim f(x) = − ∞ x→1+ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
- 34. Use the number line − .. 0 .. . + . x − 1) . . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( .+ +∞ . . ∞ . . . − − .. . f .(x) 1 . 2 . lim f(x) = + ∞ x→1− lim f(x) = − ∞ x→1+ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
- 35. Use the number line − .. 0 .. . + . x − 1) . . ( small 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( .+ +∞ . . ∞ . . ∞ . . . − − − . f .(x) 1 . 2 . lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ x→1+ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
- 36. Use the number line − .. 0 .. . + . x − 1) . . ( small 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( .+ +∞ . . ∞ . . ∞ . . ∞ . − − − + . f .(x) 1 . 2 . lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
- 37. Use the number line − .. 0 .. . + . x − 1) . . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( .+ +∞ . . ∞ . . ∞ . . ∞ . − − − + . + f .(x) 1 . 2 . lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
- 38. In English, now To explain the limit, you can say: “As x → 1− , the numerator approaches 3, and the denominator approaches 0 while remaining positive. So the limit is +∞.” . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 13 / 38
- 39. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
- 40. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
- 41. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
- 42. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
- 43. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
- 44. Limit Laws (?) with infinite limits Fact The sum of two positive or two negative infinite limits is infinite. If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then lim (f(x) + g(x)) = −∞. x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 15 / 38
- 45. Rules of Thumb with infinite limits ∞ . +∞=∞ Fact The sum of two positive or two negative infinite limits is infinite. . If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then lim (f(x) + g(x)) = −∞. x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 15 / 38
- 46. Rules of Thumb with infinite limits ∞ . +∞=∞ − . ∞ + (−∞) = −∞ Fact The sum of two positive or two negative infinite limits is infinite. . If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then lim (f(x) + g(x)) = −∞. .x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 15 / 38
- 47. Rules of Thumb with infinite limits ∞ . +∞=∞ − . ∞ + (−∞) = −∞ Fact The sum of two positive or two negative infinite limits is infinite. . If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then lim (f(x) + g(x)) = −∞. .x→a x→a x→a Remark We don’t say anything here about limits of the form ∞ − ∞. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 15 / 38
- 48. Rules of Thumb with infinite limits Kids, don't try this at home! Fact The sum of a finite limit and an infinite limit is infinite. If lim f(x) = L and lim g(x) = ±∞, then lim (f(x) + g(x)) = ±∞. x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 16 / 38
- 49. Rules of Thumb with infinite limits Kids, don't try this at home! Fact The sum of a finite limit and an infinite limit is infinite. . If lim f(x) = L and lim g(x) = ±∞, then lim (f(x) + g(x)) = ±∞. x→a x→a x→a L+∞=∞ . L − ∞ = −∞ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 16 / 38
- 50. Rules of Thumb with infinite limits Kids, don't try this at home! Fact The product of a finite limit and an infinite limit is infinite if the finite limit is not 0. If lim f(x) = L, lim g(x) = ∞, and L > 0, then lim f(x) · g(x) = ∞. x→a x→a x→a If lim f(x) = L, lim g(x) = ∞, and L < 0, then lim f(x) · g(x) = −∞. x→a x→a x→a If lim f(x) = L, lim g(x) = −∞, and L > 0, then x→a x→a lim f(x) · g(x) = −∞. x→a If lim f(x) = L, lim g(x) = −∞, and L < 0, then lim f(x) · g(x) = ∞. x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 17 / 38
- 51. Rules of Thumb with infinite limits Kids, don't try this at home! { ∞ if L > 0 . ·∞= L −∞ if L < 0. Fact The product of a finite limit and an infinite limit is infinite if the finite limit is not 0. . If lim f(x) = L, lim g(x) = ∞, and L > 0, then lim f(x) · g(x) = ∞. x→a x→a x→a If lim f(x) = L, lim g(x) = ∞, and L < 0, then lim f(x) · g(x) = −∞. x→a x→a x→a If lim f(x) = L, lim g(x) = −∞, and L > 0, then x→a x→a lim f(x) · g(x) = −∞. x→a If lim f(x) = L, lim g(x) = −∞, and L < 0, then lim f(x) · g(x) = ∞. x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 17 / 38
- 52. Rules of Thumb with infinite limits Kids, don't try this at home! { ∞ if L > 0 . ·∞= L −∞ if L < 0. Fact The product of a finite limit and an infinite limit is infinite if the finite limit is not 0. . If lim f(x) = L, lim g(x) = ∞, and L > 0, then lim f(x) · g(x) = ∞. x→a x→a x→a If lim f(x) = L, lim g(x) = ∞, and L < 0, then lim f(x) · g(x) = −∞. x→a x→a { x→a If lim f(x) =(−∞) = −∞−∞,Land0L > 0, then . · L L, lim g(x) = if > x→a x→a lim f(x) · g(x) = −∞. ∞ if L < 0. x→a If lim f(x) = L, lim g(x) = −∞, and L < 0, then lim f(x) · g(x) = ∞. x→a x→a x→a . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 17 / 38
- 53. Multiplying infinite limits Kids, don't try this at home! Fact The product of two infinite limits is infinite. If lim f(x) = ∞ and lim g(x) = ∞, then lim f(x) · g(x) = ∞. x→a x→a x→a If lim f(x) = ∞ and lim g(x) = −∞, then lim f(x) · g(x) = −∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then lim f(x) · g(x) = ∞. x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 18 / 38
- 54. Multiplying infinite limits Kids, don't try this at home! ∞·∞=∞ . ∞ · (−∞) = −∞ (−∞) · (−∞) = ∞ Fact The product of two infinite limits is infinite. . If lim f(x) = ∞ and lim g(x) = ∞, then lim f(x) · g(x) = ∞. x→a x→a x→a If lim f(x) = ∞ and lim g(x) = −∞, then lim f(x) · g(x) = −∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then lim f(x) · g(x) = ∞. x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 18 / 38
- 55. Dividing by Infinity Kids, don't try this at home! Fact The quotient of a finite limit by an infinite limit is zero. f(x) If lim f(x) = L and lim g(x) = ±∞, then lim = 0. x→a x→a x→a g(x) . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 19 / 38
- 56. Dividing by Infinity Kids, don't try this at home! Fact The quotient of a finite limit by an infinite limit is zero. f(x) If lim f(x) = L and lim g(x) = ±∞, then lim = 0. x→a x→a x→a g(x) . L . =0 ∞ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 19 / 38
- 57. Dividing by zero is still not allowed 1 . . =∞ 0 There are examples of such limit forms where the limit is ∞, −∞, undecided between the two, or truly neither. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 20 / 38
- 58. Indeterminate Limit forms L Limits of the form are indeterminate. There is no rule for evaluating 0 such a form; the limit must be examined more closely. Consider these: 1 −1 lim =∞ lim = −∞ x→0 x2 x→0 x2 1 1 lim+ = ∞ lim = −∞ x→0 x x→0− x 1 L Worst, lim is of the form , but the limit does not exist, even x→0 x sin(1/x) 0 in the left- or right-hand sense. There are infinitely many vertical asymptotes arbitrarily close to 0! . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 21 / 38
- 59. Indeterminate Limit forms Limits of the form 0 · ∞ and ∞ − ∞ are also indeterminate. Example 1 The limit lim+ sin x · is of the form 0 · ∞, but the answer is 1. x→0 x 1 The limit lim+ sin2 x · is of the form 0 · ∞, but the answer is 0. x→0 x 1 The limit lim+ sin x · 2 is of the form 0 · ∞, but the answer is ∞. x→0 x Limits of indeterminate forms may or may not “exist.” It will depend on the context. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 22 / 38
- 60. Indeterminate forms are like Tug Of War Which side wins depends on which side is stronger. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 23 / 38
- 61. Outline Infinite Limits Vertical Asymptotes Infinite Limits we Know Limit “Laws” with Infinite Limits Indeterminate Limit forms Limits at ∞ Algebraic rates of growth Rationalizing to get a limit . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 24 / 38
- 62. 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. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 25 / 38
- 63. 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→−∞ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 25 / 38
- 64. 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! . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 25 / 38
- 65. Basic limits at infinity Theorem Let n be a positive integer. Then 1 lim =0 x→∞ xn 1 lim =0 x→−∞ xn . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 26 / 38
- 66. Limit laws at infinity Fact Any limit law that concerns finite limits at a finite point a is still true if the finite point is replaced by ±∞. That is, if lim f(x) = L and lim g(x) = M, then x→∞ x→∞ lim (f(x) + g(x)) = L + M x→∞ lim (f(x) − g(x)) = L − M x→∞ lim cf(x) = c · L (for any constant c) x→∞ lim f(x) · g(x) = L · M x→∞ f(x) L lim = (if M ̸= 0), etc. x→∞ g(x) M . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 27 / 38
- 67. Using the limit laws to compute limits at ∞ Example x Find lim x→∞ x2 +1 . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 28 / 38
- 68. Using the limit laws to compute limits at ∞ Example x Find lim x→∞ x2 +1 Answer The limit is 0. y . . x . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 28 / 38
- 69. Solution Solution Factor out the largest power of x from the numerator and denominator. We have x x(1) 1 1 = 2 = · x2 +1 x (1 + 1/x2 ) x 1 + 1/x2 x 1 1 1 1 lim = lim = lim · lim x→∞ x2 + 1 x→∞ x 1 + 1/x2 x→∞ x x→∞ 1 + 1/x2 1 =0· = 0. 1+0 . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 29 / 38
- 70. Using the limit laws to compute limits at ∞ Example x Find lim x→∞ x2 +1 Answer The limit is 0. y . . x . Notice that the graph does cross the asymptote, which contradicts one of the commonly held beliefs of what an asymptote is. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 30 / 38
- 71. Solution Solution Factor out the largest power of x from the numerator and denominator. We have x x(1) 1 1 = 2 = · x2 +1 x (1 + 1/x2 ) x 1 + 1/x2 x 1 1 1 1 lim = lim = lim · lim x→∞ x2 + 1 x→∞ x 1 + 1/x2 x→∞ x x→∞ 1 + 1/x2 1 =0· = 0. 1+0 Remark Had the higher power been in the numerator, the limit would have been ∞. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 31 / 38
- 72. Another Example Example Find 2x3 + 3x + 1 lim x→∞ 4x3 + 5x2 + 7 if it exists. A does not exist B 1/2 C 0 D ∞ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 32 / 38
- 73. Another Example Example Find 2x3 + 3x + 1 lim x→∞ 4x3 + 5x2 + 7 if it exists. A does not exist B 1/2 C 0 D ∞ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 32 / 38
- 74. Solution Solution Factor out the largest power of x from the numerator and denominator. We have 2x3 + 3x + 1 x3 (2 + 3/x2 + 1/x3 ) = 3 4x3 + 5x2 + 7 x (4 + 5/x + 7/x3 ) 2x3 + 3x + 1 2 + 3/x2 + 1/x3 lim = lim x→∞ 4x3 + 5x2 + 7 x→∞ 4 + 5/x + 7/x3 2+0+0 1 = = 4+0+0 2 . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 33 / 38
- 75. Solution Solution Factor out the largest power of x from the numerator and denominator. We have 2x3 + 3x + 1 x3 (2 + 3/x2 + 1/x3 ) = 3 4x3 + 5x2 + 7 x (4 + 5/x + 7/x3 ) 2x3 + 3x + 1 2 + 3/x2 + 1/x3 lim = lim x→∞ 4x3 + 5x2 + 7 x→∞ 4 + 5/x + 7/x3 2+0+0 1 = = 4+0+0 2 Upshot When finding limits of algebraic expressions at infinity, look at the highest degree terms. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 33 / 38
- 76. Still Another Example Example Find √ 3x4 + 7 lim x→∞ x2 + 3 . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 34 / 38
- 77. Still Another Example √ √ √ . 3x4 + 7 ∼ 3x4 = 3x2 Example Find √ 3x4 + 7 . lim x→∞ x2 + 3 Answer √ The limit is 3. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 34 / 38
- 78. Solution Solution √ √ 3x4 + 7 x4 (3 + 7/x4 ) lim = lim x→∞ x2 + 3 x→∞ x2 (1 + 3/x2 ) √ x2 (3 + 7/x4 ) = lim x→∞ x2 (1 + 3/x2 ) √ (3 + 7/x4 ) = lim x→∞ 1 + 3/x2 √ 3+0 √ = = 3. 1+0 . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 35 / 38
- 79. Rationalizing to get a limit Example (√ ) Compute lim 4x2 + 17 − 2x . x→∞ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 36 / 38
- 80. Rationalizing to get a limit Example (√ ) Compute lim 4x2 + 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. (√ ) (√ ) √4x2 + 17 + 2x lim 4x 2 + 17 − 2x = lim 4x 2 + 17 − 2x · √ x→∞ x→∞ 4x2 + 17 + 2x (4x2 + 17) − 4x2 = lim √ x→∞ 4x2 + 17 + 2x 17 = lim √ =0 x→∞ 4x2 + 17 + 2x . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 36 / 38
- 81. Kick it up a notch Example (√ ) Compute lim 4x2 + 17x − 2x . x→∞ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 37 / 38
- 82. Kick it up a notch Example (√ ) Compute lim 4x2 + 17x − 2x . x→∞ Solution Same trick, different answer: (√ ) lim 4x2 + 17x − 2x x→∞ (√ √ ) 4x2 + 17 + 2x = lim + 17x − 2x · √ 4x2 x→∞ 4x2 + 17x + 2x (4x2 + 17x) − 4x2 = lim √ x→∞ 4x2 + 17x + 2x 17x 17 17 = lim √ = lim √ = x→∞ 4x2 + 17x + 2x x→∞ 4 + 17/x + 2 4 . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 37 / 38
- 83. Summary Infinity is a more complicated concept than a single number. There are rules of thumb, but there are also exceptions. Take a two-pronged approach to limits involving infinity: Look at the expression to guess the limit. Use limit rules and algebra to verify it. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 38 / 38

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