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Section 3.7   Indeterminate Forms and L’Hôpital’s                  Rule                     V63.0121.041, Calculus I      ...
Announcements         Quiz 3 in recitation this         week on Sections 2.6, 2.8,         3.1, and 3.2                   ...
Objectives         Know when a limit is of         indeterminate form:                indeterminate quotients:            ...
Experiments with funny limits           sin2 x       lim       x→0   x                                                    ...
Experiments with funny limits           sin2 x       lim        =0       x→0   x                                          ...
Experiments with funny limits           sin2 x       lim        =0       x→0   x             x       lim       x→0 sin2 x ...
Experiments with funny limits           sin2 x       lim        =0       x→0   x             x       lim        does not e...
Experiments with funny limits           sin2 x       lim         =0       x→0    x              x       lim         does n...
Experiments with funny limits           sin2 x       lim         =0       x→0    x              x       lim         does n...
Experiments with funny limits           sin2 x       lim         =0       x→0    x              x       lim         does n...
Experiments with funny limits           sin2 x       lim         =0       x→0    x              x       lim         does n...
Experiments with funny limits           sin2 x       lim         =0       x→0    x              x       lim         does n...
RecallRecall the limit laws from Chapter 2.      Limit of a sum is the sum of the limits                                  ...
RecallRecall the limit laws from Chapter 2.      Limit of a sum is the sum of the limits      Limit of a difference is the...
RecallRecall the limit laws from Chapter 2.      Limit of a sum is the sum of the limits      Limit of a difference is the...
RecallRecall the limit laws from Chapter 2.      Limit of a sum is the sum of the limits      Limit of a difference is the...
More about dividing limits      We know dividing by zero is bad.      Most of the time, if an expression’s numerator appro...
More about dividing limits      We know dividing by zero is bad.      Most of the time, if an expression’s numerator appro...
More about dividing limits      We know dividing by zero is bad.      Most of the time, if an expression’s numerator appro...
Language NoteIt depends on what the meaning of the word “is" is        Be careful with the        language here. We are no...
Indeterminate forms are like Tug Of WarWhich side wins depends on which side is stronger.                                 ...
OutlineL’Hôpital’s RuleRelative Rates of GrowthOther Indeterminate Limits   Indeterminate Products   Indeterminate Differe...
The Linear CaseQuestionIf f and g are lines and f(a) = g(a) = 0, what is                                           f(x)   ...
The Linear CaseQuestionIf f and g are lines and f(a) = g(a) = 0, what is                                           f(x)   ...
The Linear Case, Illustrated                          y                          .                                        ...
What then?      But what if the functions aren’t linear?                                                                 ....
What then?      But what if the functions aren’t linear?      Can we approximate a function near a point with a linear fun...
What then?      But what if the functions aren’t linear?      Can we approximate a function near a point with a linear fun...
What then?      But what if the functions aren’t linear?      Can we approximate a function near a point with a linear fun...
Theorem of the DayTheorem (L’Hopital’s Rule)Suppose f and g are differentiable functions and g′ (x) ̸= 0 near a(except pos...
Meet the Mathematician: LH_pital       wanted to be a military       man, but poor eyesight       forced him into math    ...
Revisiting the previous examplesExample                                  sin2 x                              lim          ...
Revisiting the previous examplesExample                                  sin2 x H     2 sin x cos x                       ...
Revisiting the previous examplesExample                                                               . in x → 0          ...
Revisiting the previous examplesExample                                                              . in x → 0           ...
Revisiting the previous examplesExample                                                              . in x → 0           ...
Revisiting the previous examplesExample                                  sin2 x H     2 sin x cos x                       ...
Revisiting the previous examplesExample                                  sin2 x H     2 sin x cos x                       ...
Revisiting the previous examplesExample                                  sin2 x H     2 sin x cos x                       ...
Revisiting the previous examplesExample                                  sin2 x H     2 sin x cos x                       ...
Revisiting the previous examplesExample                                  sin2 x H     2 sin x cos x                       ...
Revisiting the previous examplesExample                                  sin2 x H     2 sin x cos x                       ...
Revisiting the previous examplesExample                                  sin2 x H     2 sin x cos x                       ...
Revisiting the previous examplesExample                                  sin2 x H     2 sin x cos x                       ...
Revisiting the previous examplesExample                                  sin2 x H     2 sin x cos x                       ...
Revisiting the previous examplesExample                                  sin2 x H     2 sin x cos x                       ...
Revisiting the previous examplesExample                                  sin2 x H     2 sin x cos x                       ...
Another ExampleExampleFind                                             x                                        lim       ...
Beware of Red HerringsExampleFind                                             x                                        lim...
OutlineL’Hôpital’s RuleRelative Rates of GrowthOther Indeterminate Limits   Indeterminate Products   Indeterminate Differe...
Limits of Rational Functions revisitedExample           5x2 + 3x − 1Find lim                 if it exists.       x→∞ 3x2 +...
Limits of Rational Functions revisitedExample           5x2 + 3x − 1Find lim                 if it exists.       x→∞ 3x2 +...
Limits of Rational Functions revisitedExample           5x2 + 3x − 1Find lim                 if it exists.       x→∞ 3x2 +...
Limits of Rational Functions revisitedExample           5x2 + 3x − 1Find lim                 if it exists.       x→∞ 3x2 +...
Limits of Rational Functions revisitedExample           5x2 + 3x − 1Find lim                 if it exists.       x→∞ 3x2 +...
Limits of Rational Functions revisited IIExample           5x2 + 3x − 1Find lim                if it exists.       x→∞   7...
Limits of Rational Functions revisited IIExample           5x2 + 3x − 1Find lim                if it exists.       x→∞   7...
Limits of Rational Functions revisited IIExample           5x2 + 3x − 1Find lim                if it exists.       x→∞   7...
Limits of Rational Functions revisited IIExample           5x2 + 3x − 1Find lim                if it exists.       x→∞   7...
Limits of Rational Functions revisited IIIExample                   4x + 7Find lim                     if it exists.      ...
Limits of Rational Functions revisited IIIExample                   4x + 7Find lim                     if it exists.      ...
Limits of Rational Functions revisited IIIExample                   4x + 7Find lim                     if it exists.      ...
Limits of Rational Functions revisited IIIExample                   4x + 7Find lim                     if it exists.      ...
Limits of Rational FunctionsFactLet f(x) and g(x) be polynomials of degree p and q.                        f(x)     If p  ...
Exponential versus geometric growthExample           exFind lim      , if it exists.       x→∞ x2                         ...
Exponential versus geometric growthExample           exFind lim      , if it exists.       x→∞ x2SolutionWe have          ...
Exponential versus geometric growthExample           exFind lim      , if it exists.       x→∞ x2SolutionWe have          ...
Exponential versus geometric growthExample           exFind lim      , if it exists.       x→∞ x2SolutionWe have          ...
Exponential versus fractional powersExample         exFind lim √ , if it exists.    x→∞   x                               ...
Exponential versus fractional powersExample         exFind lim √ , if it exists.    x→∞   xSolution (without L’Hôpital)We ...
Exponential versus fractional powersExample         exFind lim √ , if it exists.    x→∞   xSolution (without L’Hôpital)We ...
Exponential versus any powerTheoremLet r be any positive number. Then                                       ex            ...
Exponential versus any powerTheoremLet r be any positive number. Then                                             ex      ...
Any exponential versus any powerTheoremLet a  1 and r  0. Then                                       ax                   ...
Any exponential versus any powerTheoremLet a  1 and r  0. Then                                           ax               ...
Any exponential versus any powerTheoremLet a  1 and r  0. Then                                           ax               ...
Logarithmic versus power growthTheoremLet r be any positive number. Then                                      ln x        ...
Logarithmic versus power growthTheoremLet r be any positive number. Then                                      ln x        ...
OutlineL’Hôpital’s RuleRelative Rates of GrowthOther Indeterminate Limits   Indeterminate Products   Indeterminate Differe...
Indeterminate productsExampleFind                                          √                                      lim+    ...
Indeterminate productsExampleFind                                                    √                                    ...
Indeterminate productsExampleFind                                           √                                       lim+  ...
Indeterminate productsExampleFind                                           √                                       lim+  ...
Indeterminate productsExampleFind                                           √                                       lim+  ...
Indeterminate productsExampleFind                                           √                                       lim+  ...
Indeterminate differencesExample                                         (             )                                  ...
Indeterminate differencesExample                                         (             )                                  ...
Indeterminate differencesExample                                         (             )                                  ...
Indeterminate differencesExample                                         (             )                                  ...
Indeterminate differencesExample                                         (             )                                  ...
Checking your work                                  .                                      tan 2x                         ...
Indeterminate powersExampleFind lim+ (1 − 2x)1/x       x→0                                                                ...
Indeterminate powersExampleFind lim+ (1 − 2x)1/x       x→0Take the logarithm:      (                 )          (         ...
Indeterminate powersExampleFind lim+ (1 − 2x)1/x       x→0Take the logarithm:      (                 )          (         ...
Indeterminate powersExampleFind lim+ (1 − 2x)1/x       x→0Take the logarithm:      (                 )          (         ...
Another indeterminate power limitExample                                       lim (3x)4x                                 ...
Another indeterminate power limitExample                                        lim (3x)4x                                ...
Summary Form           Method    0    0           L’Hôpital’s rule directly    ∞    ∞           L’Hôpital’s rule directly0...
Final Thoughts      L’Hôpital’s Rule only works on indeterminate quotients      Luckily, most indeterminate limits can be ...
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Lesson 17: Indeterminate Forms and L'Hopital's Rule (Section 041 slides)

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L'Hopital's Rule allows us to resolve limits of indeterminate form: 0/0, infinity/infinity, infinity-infinity, 0^0, 1^infinity, and infinity^0

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Transcript of "Lesson 17: Indeterminate Forms and L'Hopital's Rule (Section 041 slides)"

  1. 1. Section 3.7 Indeterminate Forms and L’Hôpital’s Rule V63.0121.041, Calculus I New York University November 3, 2010Announcements Quiz 3 in recitation this week on Sections 2.6, 2.8, 3.1, and 3.2 . . . . . .
  2. 2. Announcements Quiz 3 in recitation this week on Sections 2.6, 2.8, 3.1, and 3.2 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 2 / 34
  3. 3. Objectives Know when a limit is of indeterminate form: indeterminate quotients: 0/0, ∞/∞ indeterminate products: 0×∞ indeterminate differences: ∞ − ∞ indeterminate powers: 00 , ∞0 , and 1∞ Resolve limits in indeterminate form . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 3 / 34
  4. 4. Experiments with funny limits sin2 x lim x→0 x . . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
  5. 5. Experiments with funny limits sin2 x lim =0 x→0 x . . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
  6. 6. Experiments with funny limits sin2 x lim =0 x→0 x x lim x→0 sin2 x . . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
  7. 7. Experiments with funny limits sin2 x lim =0 x→0 x x lim does not exist x→0 sin2 x . . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
  8. 8. Experiments with funny limits sin2 x lim =0 x→0 x x lim does not exist x→0 sin2 x . sin2 x lim x→0 sin(x2 ) . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
  9. 9. Experiments with funny limits sin2 x lim =0 x→0 x x lim does not exist x→0 sin2 x . sin2 x lim =1 x→0 sin(x2 ) . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
  10. 10. Experiments with funny limits sin2 x lim =0 x→0 x x lim does not exist x→0 sin2 x . sin2 x lim =1 x→0 sin(x2 ) sin 3x lim x→0 sin x . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
  11. 11. Experiments with funny limits sin2 x lim =0 x→0 x x lim does not exist x→0 sin2 x . sin2 x lim =1 x→0 sin(x2 ) sin 3x lim =3 x→0 sin x . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
  12. 12. Experiments with funny limits sin2 x lim =0 x→0 x x lim does not exist x→0 sin2 x . sin2 x lim =1 x→0 sin(x2 ) sin 3x lim =3 x→0 sin x 0All of these are of the form , and since we can get different answers 0in different cases, we say this form is indeterminate. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
  13. 13. RecallRecall the limit laws from Chapter 2. Limit of a sum is the sum of the limits . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 5 / 34
  14. 14. RecallRecall the limit laws from Chapter 2. Limit of a sum is the sum of the limits Limit of a difference is the difference of the limits . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 5 / 34
  15. 15. RecallRecall the limit laws from Chapter 2. Limit of a sum is the sum of the limits Limit of a difference is the difference of the limits Limit of a product is the product of the limits . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 5 / 34
  16. 16. RecallRecall the limit laws from Chapter 2. Limit of a sum is the sum of the limits Limit of a difference is the difference of the limits Limit of a product is the product of the limits Limit of a quotient is the quotient of the limits ... whoops! This is true as long as you don’t try to divide by zero. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 5 / 34
  17. 17. More about dividing limits We know dividing by zero is bad. Most of the time, if an expression’s numerator approaches a finite number and denominator approaches zero, the quotient approaches some kind of infinity. For example: 1 cos x lim+ = +∞ lim = −∞ x→0 x x→0 − x3 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 6 / 34
  18. 18. More about dividing limits We know dividing by zero is bad. Most of the time, if an expression’s numerator approaches a finite number and denominator approaches zero, the quotient approaches some kind of infinity. For example: 1 cos x lim+ = +∞ lim = −∞ x→0 x x→0 − x3 An exception would be something like 1 lim = lim x csc x. x→∞ 1 sin x x→∞ x which does not exist and is not infinite. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 6 / 34
  19. 19. More about dividing limits We know dividing by zero is bad. Most of the time, if an expression’s numerator approaches a finite number and denominator approaches zero, the quotient approaches some kind of infinity. For example: 1 cos x lim+ = +∞ lim = −∞ x→0 x x→0 − x3 An exception would be something like 1 lim = lim x csc x. x→∞ 1 sin x x→∞ x which does not exist and is not infinite. Even less predictable: numerator and denominator both go to zero. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 6 / 34
  20. 20. Language NoteIt depends on what the meaning of the word “is" is Be careful with the language here. We are not saying that the limit in each 0 case “is” , and therefore 0 nonexistent because this expression is undefined. 0 The limit is of the form , 0 which means we cannot evaluate it with our limit laws. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 7 / 34
  21. 21. Indeterminate forms are like Tug Of WarWhich side wins depends on which side is stronger. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 8 / 34
  22. 22. OutlineL’Hôpital’s RuleRelative Rates of GrowthOther Indeterminate Limits Indeterminate Products Indeterminate Differences Indeterminate Powers . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 9 / 34
  23. 23. The Linear CaseQuestionIf f and g are lines and f(a) = g(a) = 0, what is f(x) lim ? x→a g(x) . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 10 / 34
  24. 24. The Linear CaseQuestionIf f and g are lines and f(a) = g(a) = 0, what is f(x) lim ? x→a g(x)SolutionThe functions f and g can be written in the form f(x) = m1 (x − a) g(x) = m2 (x − a)So f(x) m = 1 g(x) m2 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 10 / 34
  25. 25. The Linear Case, Illustrated y . y . = g(x) y . = f(x) g . (x) a . f .(x) . . . x . x . f(x) f(x) − f(a) (f(x) − f(a))/(x − a) m = = = 1 g(x) g(x) − g(a) (g(x) − g(a))/(x − a) m2 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 11 / 34
  26. 26. What then? But what if the functions aren’t linear? . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 12 / 34
  27. 27. What then? But what if the functions aren’t linear? Can we approximate a function near a point with a linear function? . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 12 / 34
  28. 28. What then? But what if the functions aren’t linear? Can we approximate a function near a point with a linear function? What would be the slope of that linear function? . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 12 / 34
  29. 29. What then? But what if the functions aren’t linear? Can we approximate a function near a point with a linear function? What would be the slope of that linear function? The derivative! . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 12 / 34
  30. 30. Theorem of the DayTheorem (L’Hopital’s Rule)Suppose f and g are differentiable functions and g′ (x) ̸= 0 near a(except possibly at a). Suppose that lim f(x) = 0 and lim g(x) = 0 x→a x→aor lim f(x) = ±∞ and lim g(x) = ±∞ x→a x→aThen f(x) f′ (x) lim = lim ′ , x→a g(x) x→a g (x)if the limit on the right-hand side is finite, ∞, or −∞. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 13 / 34
  31. 31. Meet the Mathematician: LH_pital wanted to be a military man, but poor eyesight forced him into math did some math on his own (solved the “brachistocrone problem”) paid a stipend to Johann Bernoulli, who proved this theorem and named it after him! Guillaume François Antoine, Marquis de L’Hôpital (French, 1661–1704) . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 14 / 34
  32. 32. Revisiting the previous examplesExample sin2 x lim x→0 x . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  33. 33. Revisiting the previous examplesExample sin2 x H 2 sin x cos x lim = lim x→0 x x→0 1 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  34. 34. Revisiting the previous examplesExample . in x → 0 s . sin2 x H 2 sin x cos x lim = lim x→0 x x→0 1 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  35. 35. Revisiting the previous examplesExample . in x → 0 s . sin2 x H 2 sin x cos x lim = lim =0 x→0 x x→0 1 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  36. 36. Revisiting the previous examplesExample . in x → 0 s . sin2 x H 2 sin x cos x lim = lim =0 x→0 x x→0 1Example sin2 x lim x→0 sin x2 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  37. 37. Revisiting the previous examplesExample sin2 x H 2 sin x cos x lim = lim =0 x→0 x x→0 1Example . umerator → 0 n . sin2 x lim x→0 sin x2 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  38. 38. Revisiting the previous examplesExample sin2 x H 2 sin x cos x lim = lim =0 x→0 x x→0 1Example . umerator → 0 n . sin2 x lim . x→0 sin x2 . enominator → 0 d . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  39. 39. Revisiting the previous examplesExample sin2 x H 2 sin x cos x lim = lim =0 x→0 x x→0 1Example . umerator → 0 n . sin2 x H sin x cos x 2 lim 2. = lim ( ) x→0 sin x x→0 cos x2 (x) 2 . enominator → 0 d . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  40. 40. Revisiting the previous examplesExample sin2 x H 2 sin x cos x lim = lim =0 x→0 x x→0 1Example . umerator → 0 n . sin2 x H sin x cos x 2 lim = lim ( ) x→0 sin x2 x→0 cos x2 (x) 2 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  41. 41. Revisiting the previous examplesExample sin2 x H 2 sin x cos x lim = lim =0 x→0 x x→0 1Example . umerator → 0 n . sin2 x H sin x cos x 2 lim = lim ( ) . x→0 sin x2 x→0 cos x2 (x ) 2 . enominator → 0 d . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  42. 42. Revisiting the previous examplesExample sin2 x H 2 sin x cos x lim = lim =0 x→0 x x→0 1Example . umerator → 0 n . sin2 x H sin x cos x H 2 cos2 x − sin2 x lim = lim ( ) . = lim x→0 sin x2 x→0 cos x2 (x ) 2 x→0 cos x2 − 2x2 sin(x2 ) . enominator → 0 d . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  43. 43. Revisiting the previous examplesExample sin2 x H 2 sin x cos x lim = lim =0 x→0 x x→0 1Example . umerator → 1 n . sin2 x H sin x cos x H 2 cos2 x − sin2 x lim = lim ( ) = lim x→0 sin x2 x→0 cos x2 (x) 2 x→0 cos x2 − 2x2 sin(x2 ) . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  44. 44. Revisiting the previous examplesExample sin2 x H 2 sin x cos x lim = lim =0 x→0 x x→0 1Example . umerator → 1 n . sin2 x H sin x cos x H 2 cos2 x − sin2 x lim = lim ( ) = lim . x→0 sin x2 x→0 cos x2 (x) 2 x→0 cos x2 − 2x2 sin(x2 ) . enominator → 1 d . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  45. 45. Revisiting the previous examplesExample sin2 x H 2 sin x cos x lim = lim =0 x→0 x x→0 1Example sin2 x H sin x cos x H 2 cos2 x − sin2 x lim = lim ( ) = lim =1 x→0 sin x2 x→0 cos x2 (x) 2 x→0 cos x2 − 2x2 sin(x2 ) . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  46. 46. Revisiting the previous examplesExample sin2 x H 2 sin x cos x lim = lim =0 x→0 x x→0 1Example sin2 x H sin x cos x H 2 cos2 x − sin2 x lim = lim ( ) = lim =1 x→0 sin x2 x→0 cos x2 (x) 2 x→0 cos x2 − 2x2 sin(x2 )Example sin 3x lim x→0 sin x . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  47. 47. Revisiting the previous examplesExample sin2 x H 2 sin x cos x lim = lim =0 x→0 x x→0 1Example sin2 x H sin x cos x H 2 cos2 x − sin2 x lim = lim ( ) = lim =1 x→0 sin x2 x→0 cos x2 (x) 2 x→0 cos x2 − 2x2 sin(x2 )Example sin 3x H 3 cos 3x lim = lim = 3. x→0 sin x x→0 cos x . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
  48. 48. Another ExampleExampleFind x lim x→0 cos x . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 16 / 34
  49. 49. Beware of Red HerringsExampleFind x lim x→0 cos xSolutionThe limit of the denominator is 1, not 0, so L’Hôpital’s rule does notapply. The limit is 0. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 16 / 34
  50. 50. OutlineL’Hôpital’s RuleRelative Rates of GrowthOther Indeterminate Limits Indeterminate Products Indeterminate Differences Indeterminate Powers . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 17 / 34
  51. 51. Limits of Rational Functions revisitedExample 5x2 + 3x − 1Find lim if it exists. x→∞ 3x2 + 7x + 27 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 18 / 34
  52. 52. Limits of Rational Functions revisitedExample 5x2 + 3x − 1Find lim if it exists. x→∞ 3x2 + 7x + 27SolutionUsing L’Hôpital: 5x2 + 3x − 1 lim x→∞ 3x2 + 7x + 27 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 18 / 34
  53. 53. Limits of Rational Functions revisitedExample 5x2 + 3x − 1Find lim if it exists. x→∞ 3x2 + 7x + 27SolutionUsing L’Hôpital: 5x2 + 3x − 1 H 10x + 3 lim = lim x→∞ 3x2 + 7x + 27 x→∞ 6x + 7 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 18 / 34
  54. 54. Limits of Rational Functions revisitedExample 5x2 + 3x − 1Find lim if it exists. x→∞ 3x2 + 7x + 27SolutionUsing L’Hôpital: 5x2 + 3x − 1 H 10x + 3 H 10 5 lim = lim = lim = x→∞ 3x2 + 7x + 27 x→∞ 6x + 7 x→∞ 6 3 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 18 / 34
  55. 55. Limits of Rational Functions revisitedExample 5x2 + 3x − 1Find lim if it exists. x→∞ 3x2 + 7x + 27SolutionUsing L’Hôpital: 5x2 + 3x − 1 H 10x + 3 H 10 5 lim = lim = lim = x→∞ 3x2 + 7x + 27 x→∞ 6x + 7 x→∞ 6 3 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 18 / 34
  56. 56. Limits of Rational Functions revisited IIExample 5x2 + 3x − 1Find lim if it exists. x→∞ 7x + 27 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 19 / 34
  57. 57. Limits of Rational Functions revisited IIExample 5x2 + 3x − 1Find lim if it exists. x→∞ 7x + 27SolutionUsing L’Hôpital: 5x2 + 3x − 1 lim x→∞ 7x + 27 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 19 / 34
  58. 58. Limits of Rational Functions revisited IIExample 5x2 + 3x − 1Find lim if it exists. x→∞ 7x + 27SolutionUsing L’Hôpital: 5x2 + 3x − 1 H 10x + 3 lim = lim x→∞ 7x + 27 x→∞ 7 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 19 / 34
  59. 59. Limits of Rational Functions revisited IIExample 5x2 + 3x − 1Find lim if it exists. x→∞ 7x + 27SolutionUsing L’Hôpital: 5x2 + 3x − 1 H 10x + 3 lim = lim =∞ x→∞ 7x + 27 x→∞ 7 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 19 / 34
  60. 60. Limits of Rational Functions revisited IIIExample 4x + 7Find lim if it exists. x→∞ 3x2 + 7x + 27 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 20 / 34
  61. 61. Limits of Rational Functions revisited IIIExample 4x + 7Find lim if it exists. x→∞ 3x2 + 7x + 27SolutionUsing L’Hôpital: 4x + 7 lim x→∞ 3x2 + 7x + 27 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 20 / 34
  62. 62. Limits of Rational Functions revisited IIIExample 4x + 7Find lim if it exists. x→∞ 3x2 + 7x + 27SolutionUsing L’Hôpital: 4x + 7 H 4 lim = lim x→∞ 3x2 + 7x + 27 x→∞ 6x + 7 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 20 / 34
  63. 63. Limits of Rational Functions revisited IIIExample 4x + 7Find lim if it exists. x→∞ 3x2 + 7x + 27SolutionUsing L’Hôpital: 4x + 7 H 4 lim = lim =0 x→∞ 3x2 + 7x + 27 x→∞ 6x + 7 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 20 / 34
  64. 64. Limits of Rational FunctionsFactLet f(x) and g(x) be polynomials of degree p and q. f(x) If p q, then lim =∞ x→∞ g(x) f(x) If p q, then lim =0 x→∞ g(x) f(x) If p = q, then lim is the ratio of the leading coefficients of f x→∞ g(x) and g. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 21 / 34
  65. 65. Exponential versus geometric growthExample exFind lim , if it exists. x→∞ x2 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 22 / 34
  66. 66. Exponential versus geometric growthExample exFind lim , if it exists. x→∞ x2SolutionWe have ex H ex H ex lim = lim = lim = ∞. x→∞ x2 x→∞ 2x x→∞ 2 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 22 / 34
  67. 67. Exponential versus geometric growthExample exFind lim , if it exists. x→∞ x2SolutionWe have ex H ex H ex lim = lim = lim = ∞. x→∞ x2 x→∞ 2x x→∞ 2Example exWhat about lim ? x→∞ x3 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 22 / 34
  68. 68. Exponential versus geometric growthExample exFind lim , if it exists. x→∞ x2SolutionWe have ex H ex H ex lim = lim = lim = ∞. x→∞ x2 x→∞ 2x x→∞ 2Example exWhat about lim ? x→∞ x3AnswerStill ∞. (Why?) . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 22 / 34
  69. 69. Exponential versus fractional powersExample exFind lim √ , if it exists. x→∞ x . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 23 / 34
  70. 70. Exponential versus fractional powersExample exFind lim √ , if it exists. x→∞ xSolution (without L’Hôpital)We have for all x 1, x1/2 x1 , so ex ex x1/2 xThe right hand side tends to ∞, so the left-hand side must, too. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 23 / 34
  71. 71. Exponential versus fractional powersExample exFind lim √ , if it exists. x→∞ xSolution (without L’Hôpital)We have for all x 1, x1/2 x1 , so ex ex x1/2 xThe right hand side tends to ∞, so the left-hand side must, too.Solution (with L’Hôpital) ex ex √ lim √ = lim 1 = lim 2 xex = ∞ x→∞ x x→∞ 2 x−1/2 x→∞ . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 23 / 34
  72. 72. Exponential versus any powerTheoremLet r be any positive number. Then ex lim = ∞. x→∞ xr . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 24 / 34
  73. 73. Exponential versus any powerTheoremLet r be any positive number. Then ex lim = ∞. x→∞ xrProof.If r is a positive integer, then apply L’Hôpital’s rule r times to thefraction. You get ex H H ex lim = . . . = lim = ∞. x→∞ xr x→∞ r!If r is not an integer, let m be the smallest integer greater than r. Then ex exif x 1, xr xm , so r m . The right-hand side tends to ∞ by the x xprevious step. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 24 / 34
  74. 74. Any exponential versus any powerTheoremLet a 1 and r 0. Then ax lim = ∞. x→∞ xr . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 25 / 34
  75. 75. Any exponential versus any powerTheoremLet a 1 and r 0. Then ax lim = ∞. x→∞ xrProof.If r is a positive integer, we have ax H H (ln a)r ax lim = . . . = lim = ∞. x→∞ xr x→∞ r!If r isn’t an integer, we can compare it as before. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 25 / 34
  76. 76. Any exponential versus any powerTheoremLet a 1 and r 0. Then ax lim = ∞. x→∞ xrProof.If r is a positive integer, we have ax H H (ln a)r ax lim = . . . = lim = ∞. x→∞ xr x→∞ r!If r isn’t an integer, we can compare it as before. (1.00000001)xSo even lim = ∞! x→∞ x100000000 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 25 / 34
  77. 77. Logarithmic versus power growthTheoremLet r be any positive number. Then ln x lim = 0. x→∞ xr . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 26 / 34
  78. 78. Logarithmic versus power growthTheoremLet r be any positive number. Then ln x lim = 0. x→∞ xrProof.One application of L’Hôpital’s Rule here suffices: ln x H 1/x 1 limr = lim r−1 = lim r = 0. x→∞ x x→∞ rx x→∞ rx . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 26 / 34
  79. 79. OutlineL’Hôpital’s RuleRelative Rates of GrowthOther Indeterminate Limits Indeterminate Products Indeterminate Differences Indeterminate Powers . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 27 / 34
  80. 80. Indeterminate productsExampleFind √ lim+ x ln x x→0This limit is of the form 0 · (−∞). . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 28 / 34
  81. 81. Indeterminate productsExampleFind √ lim+ x ln x x→0This limit is of the form 0 · (−∞).SolutionJury-rig the expression to make an indeterminate quotient. Then applyL’Hôpital’s Rule: √ lim x ln x x→0+ . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 28 / 34
  82. 82. Indeterminate productsExampleFind √ lim+ x ln x x→0This limit is of the form 0 · (−∞).SolutionJury-rig the expression to make an indeterminate quotient. Then applyL’Hôpital’s Rule: √ ln x lim x ln x = lim+ 1 √ x→0+ x→0 / x . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 28 / 34
  83. 83. Indeterminate productsExampleFind √ lim+ x ln x x→0This limit is of the form 0 · (−∞).SolutionJury-rig the expression to make an indeterminate quotient. Then applyL’Hôpital’s Rule: √ ln x H x−1 lim x ln x = lim+ 1 √ = lim+ 1 x→0+ x→0 / x x→0 − 2 x−3/2 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 28 / 34
  84. 84. Indeterminate productsExampleFind √ lim+ x ln x x→0This limit is of the form 0 · (−∞).SolutionJury-rig the expression to make an indeterminate quotient. Then applyL’Hôpital’s Rule: √ ln x H x−1 lim x ln x = lim+ 1 √ = lim+ 1 x→0+ x→0 / x x→0 − 2 x−3/2 √ = lim+ −2 x x→0 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 28 / 34
  85. 85. Indeterminate productsExampleFind √ lim+ x ln x x→0This limit is of the form 0 · (−∞).SolutionJury-rig the expression to make an indeterminate quotient. Then applyL’Hôpital’s Rule: √ ln x H x−1 lim x ln x = lim+ 1 √ = lim+ 1 x→0+ x→0 / x x→0 − 2 x−3/2 √ = lim+ −2 x = 0 x→0 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 28 / 34
  86. 86. Indeterminate differencesExample ( ) 1 lim+ − cot 2x x→0 xThis limit is of the form ∞ − ∞. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 29 / 34
  87. 87. Indeterminate differencesExample ( ) 1 lim+ − cot 2x x→0 xThis limit is of the form ∞ − ∞.SolutionAgain, rig it to make an indeterminate quotient. sin(2x) − x cos(2x) lim+ x→0 x sin(2x) . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 29 / 34
  88. 88. Indeterminate differencesExample ( ) 1 lim+ − cot 2x x→0 xThis limit is of the form ∞ − ∞.SolutionAgain, rig it to make an indeterminate quotient. sin(2x) − x cos(2x) H cos(2x) + 2x sin(2x) lim+ = lim+ x→0 x sin(2x) x→0 2x cos(2x) + sin(2x) . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 29 / 34
  89. 89. Indeterminate differencesExample ( ) 1 lim+ − cot 2x x→0 xThis limit is of the form ∞ − ∞.SolutionAgain, rig it to make an indeterminate quotient. sin(2x) − x cos(2x) H cos(2x) + 2x sin(2x) lim+ = lim+ x→0 x sin(2x) x→0 2x cos(2x) + sin(2x) =∞ . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 29 / 34
  90. 90. Indeterminate differencesExample ( ) 1 lim+ − cot 2x x→0 xThis limit is of the form ∞ − ∞.SolutionAgain, rig it to make an indeterminate quotient. sin(2x) − x cos(2x) H cos(2x) + 2x sin(2x) lim+ = lim+ x→0 x sin(2x) x→0 2x cos(2x) + sin(2x) =∞The limit is +∞ becuase the numerator tends to 1 while thedenominator tends to zero but remains positive. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 29 / 34
  91. 91. Checking your work . tan 2x lim = 1, so for small x, x→0 2x 1 tan 2x ≈ 2x. So cot 2x ≈ and . 2x 1 1 1 1 − cot 2x ≈ − = →∞ x x 2x 2x as x → 0+ . . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 30 / 34
  92. 92. Indeterminate powersExampleFind lim+ (1 − 2x)1/x x→0 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 31 / 34
  93. 93. Indeterminate powersExampleFind lim+ (1 − 2x)1/x x→0Take the logarithm: ( ) ( ) ln(1 − 2x) ln lim+ (1 − 2x) 1/x = lim+ ln (1 − 2x)1/x = lim+ x→0 x→0 x→0 x . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 31 / 34
  94. 94. Indeterminate powersExampleFind lim+ (1 − 2x)1/x x→0Take the logarithm: ( ) ( ) ln(1 − 2x) ln lim+ (1 − 2x) 1/x = lim+ ln (1 − 2x)1/x = lim+ x→0 x→0 x→0 x 0This limit is of the form , so we can use L’Hôpital: 0 −2 ln(1 − 2x) H 1−2x lim+ = lim+ = −2 x→0 x x→0 1 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 31 / 34
  95. 95. Indeterminate powersExampleFind lim+ (1 − 2x)1/x x→0Take the logarithm: ( ) ( ) ln(1 − 2x) ln lim+ (1 − 2x) 1/x = lim+ ln (1 − 2x)1/x = lim+ x→0 x→0 x→0 x 0This limit is of the form , so we can use L’Hôpital: 0 −2 ln(1 − 2x) H 1−2x lim+ = lim+ = −2 x→0 x x→0 1This is not the answer, it’s the log of the answer! So the answer wewant is e−2 . . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 31 / 34
  96. 96. Another indeterminate power limitExample lim (3x)4x x→0 . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 32 / 34
  97. 97. Another indeterminate power limitExample lim (3x)4x x→0Solution ln lim+ (3x)4x = lim+ ln(3x)4x = lim+ 4x ln(3x) x→0 x→0 x→0 ln(3x) H 3/3x = lim+ 1 = lim+ −1/4x2 x→0 /4x x→0 = lim+ (−4x) = 0 x→0So the answer is e0 = 1. . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 32 / 34
  98. 98. Summary Form Method 0 0 L’Hôpital’s rule directly ∞ ∞ L’Hôpital’s rule directly0·∞ jiggle to make 0 or ∞ . 0 ∞∞−∞ combine to make an indeterminate product or quotient 00 take ln to make an indeterminate product ∞0 ditto 1∞ ditto . . . . . .V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 33 / 34
  99. 99. Final Thoughts L’Hôpital’s Rule only works on indeterminate quotients Luckily, most indeterminate limits can be transformed into indeterminate quotients L’Hôpital’s Rule gives wrong answers for non-indeterminate limits! . . . . . . V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 34 / 34
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