Lesson 17: Indeterminate forms and l'Hôpital's Rule (handout)
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Notes Sec on 3.7 Indeterminate forms and lHôpital’s Rule V63.0121.001: Calculus I Professor Ma hew Leingang New York University . . . Notes Announcements Midterm has been returned. Please see FAQ on Blackboard (under ”Exams and Quizzes”) Quiz 3 this week in recita on on Sec on 2.6, 2.8, 3.1, 3.2 . . Notes Objectives Know when a limit is of indeterminate form: indeterminate quo ents: 0/0, ∞/∞ indeterminate products: 0×∞ indeterminate diﬀerences: ∞ − ∞ indeterminate powers: 00 , ∞0 , and 1∞ Resolve limits in indeterminate form . . . 1.
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Notes Recall Recall 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 quo ent is the quo ent of the limits ... whoops! This is true as long as you don’t try to divide by zero. . . Notes More about dividing limits We know dividing by zero is bad. Most of the me, if an expression’s numerator approaches a ﬁnite nonzero number and denominator approaches zero, the quo ent has an inﬁnite. For example: 1 cos x lim+ = +∞ lim− = −∞ x→0 x x→0 x3 . . Notes Why 1/0 ̸= ∞ 1 Consider the func on f(x) = 1 . x sin x y . x Then lim f(x) is of the form 1/0, but the limit does not exist and is x→∞ not inﬁnite. Even less predictable: when numerator and denominator both go to zero. . . . 2.
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Notes Experiments with funny limits sin2 x lim =0 x→0 x x lim does not exist x→0 sin2 x 2 sin x . lim 2) =1 x→0 sin(x sin 3x lim =3 x→0 sin x 0 All of these are of the form , and since we can get diﬀerent 0 answers in diﬀerent cases, we say this form is indeterminate. . . Notes Experiments with funny limits sin2 x lim =0 x→0 x x lim does not exist x→0 sin2 x 2 sin x . lim =1 x→0 sin(x2 ) sin 3x lim =3 x→0 sin x 0 All of these are of the form , and since we can get diﬀerent 0 answers in diﬀerent cases, we say this form is indeterminate. . . Notes Experiments with funny limits sin2 x lim =0 x→0 x x lim does not exist x→0 sin2 x 2 sin x . lim 2) =1 x→0 sin(x sin 3x lim =3 x→0 sin x 0 All of these are of the form , and since we can get diﬀerent 0 answers in diﬀerent cases, we say this form is indeterminate. . . . 3.
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Notes Experiments with funny limits sin2 x lim =0 x→0 x x lim does not exist x→0 sin2 x 2 sin x . lim 2) =1 x→0 sin(x sin 3x lim =3 x→0 sin x 0 All of these are of the form , and since we can get diﬀerent 0 answers in diﬀerent cases, we say this form is indeterminate. . . Language Note Notes It 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 nonexistent 0 because this expression is undeﬁned. 0 The limit is of the form , which means 0 we cannot evaluate it with our limit laws. . . Notes Indeterminate forms are like Tug Of War . Which side wins depends on which side is stronger. . . 4.
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Notes Outline L’Hôpital’s Rule Rela ve Rates of Growth Other Indeterminate Limits Indeterminate Products Indeterminate Diﬀerences Indeterminate Powers . . Notes The Linear Case Ques on f(x) If f and g are lines and f(a) = g(a) = 0, what is lim ? x→a g(x) Solu on The func ons f and g can be wri en in the form f(x) = m1 (x − a) g(x) = m2 (x − a) So f(x) m1 = g(x) m2 . . Notes The Linear Case, Illustrated y y = g(x) y = f(x) a f(x) g(x) . x x f(x) f(x) − f(a) (f(x) − f(a))/(x − a) m1 = = = g(x) g(x) − g(a) (g(x) − g(a))/(x − a) m2 . . . 5.
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Notes What then? But what if the func ons aren’t linear? Can we approximate a func on near a point with a linear func on? What would be the slope of that linear func on? . . Notes Theorem of the Day Theorem (L’Hopital’s Rule) Suppose f and g are diﬀeren able func ons and g′ (x) ̸= 0 near a (except possibly at a). Suppose that lim f(x) = 0 and lim g(x) = 0 x→a x→a or lim f(x) = ±∞ and lim g(x) = ±∞ x→a x→a Then 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 −∞. . . Notes Meet the Mathematician 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 s pend to Johann Bernoulli, who proved this Guillaume François Antoine, theorem and named it a er him! Marquis de L’Hôpital (French, 1661–1704) . . . 6.
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Notes Revisiting the previous examples Example sin2 x H 2 sin x cos x lim = lim =0 x→0 x x→0 1 . . Notes Revisiting the previous examples Example sin2 x H sin x cos x H 2 cos2 x − sin2 x lim = lim = lim =1 x→0 sin x 2 x→0 (cos x 2 ) (2x) x→0 cos x2 − 2x2 sin(x2 ) . . Notes Revisiting the previous examples Example sin 3x H 3 cos 3x lim = lim = 3. x→0 sin x x→0 cos x . . . 7.
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Notes Beware of Red Herrings Example Find x lim x→0 cos x Solu on . . Notes Outline L’Hôpital’s Rule Rela ve Rates of Growth Other Indeterminate Limits Indeterminate Products Indeterminate Diﬀerences Indeterminate Powers . . Limits of Rational Functions Notes revisited Example 5x2 + 3x − 1 Find lim if it exists. x→∞ 3x2 + 7x + 27 Solu on Using L’Hôpital: 5x2 + 3x − 1 H 10x + 3 H 10 5 lim = lim = lim = x→∞ 3x2 + 7x + 27 x→∞ 6x + 7 x→∞ 6 3 . . . 8.
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Limits of Rational Functions Notes revisited II Example 5x2 + 3x − 1 Find lim if it exists. x→∞ 7x + 27 Solu on Using L’Hôpital: 5x2 + 3x − 1 H 10x + 3 lim = lim =∞ x→∞ 7x + 27 x→∞ 7 . . Limits of Rational Functions Notes revisited III Example 4x + 7 Find lim if it exists. x→∞ 3x2 + 7x + 27 Solu on Using L’Hôpital: 4x + 7 H 4 lim = lim =0 x→∞ 3x2 + 7x + 27 x→∞ 6x + 7 . . Notes Limits of Rational Functions Fact Let 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 ra o of the leading coeﬃcients of x→∞ g(x) f and g. . . . 9.
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Notes Exponential vs. geometric growth Example ex Find lim , if it exists. x→∞ x2 Solu on We have ex H ex H ex lim = lim = lim = ∞. x→∞ x2 x→∞ 2x x→∞ 2 . . Notes Exponential vs. geometric growth Example ex What about lim ? x→∞ x3 Answer Solu on . . Notes Exponential vs. fractional powers Example ex Find lim √ , if it exists. x→∞ x Solu on (without L’Hôpital) We have for all x 1, x1/2 x1 , so ex ex 1/2 x x The right hand side tends to ∞, so the le -hand side must, too. . . . 10.
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Notes Exponential vs. fractional powers Example ex Find lim √ , if it exists. x→∞ x Solu on (with L’Hôpital) ex ex √ lim √ = lim 1 −1/2 = lim 2 xex = ∞ x→∞ x x→∞ x 2 x→∞ . . Notes Exponential vs. any power Theorem ex Let r be any posi ve number. Then lim = ∞. x→∞ xr Proof. If r is a posi ve integer, then apply L’Hôpital’s rule r mes to the frac- on. You get ex H H ex lim = . . . = lim = ∞. x→∞ xr x→∞ r! . . Notes Exponential vs. any power Theorem ex Let r be any posi ve number. Then lim = ∞. x→∞ xr Proof. If r is not an integer, let m be the smallest integer greater than r. Then ex ex if x 1, xr xm , so r m . The right-hand side tends to ∞ by the x x previous step. . . . 11.
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Notes Any exponential vs. any power Theorem ax Let a 1 and r 0. Then lim = ∞. x→∞ xr Proof. If r is a posi ve 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)x So even lim = ∞! . x→∞ x100000000 . Notes Logarithmic versus power growth Theorem ln x Let r be any posi ve number. Then lim = 0. x→∞ xr Proof. One applica on of L’Hôpital’s Rule here suﬃces: ln x H 1/x 1 lim = lim r−1 = lim r = 0. x→∞ xr x→∞ rx x→∞ rx . . Notes Outline L’Hôpital’s Rule Rela ve Rates of Growth Other Indeterminate Limits Indeterminate Products Indeterminate Diﬀerences Indeterminate Powers . . . 12.
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Notes Indeterminate products Example √ Find lim+ x ln x x→0 This limit is of the form 0 · (−∞). Solu on Jury-rig the expression to make an indeterminate quo ent. Then apply L’Hôpital’s Rule: √ ln x H x−1 √ lim x ln x = lim+ 1 √ = lim+ 1 −3/2 = lim+ −2 x = 0 x→0+ x→0 / x x→0 − 2 x x→0 . . Notes Indeterminate diﬀerences Example ( ) 1 lim − cot 2x x→0+ x This limit is of the form ∞ − ∞. . . Notes Indeterminate Diﬀerences Solu on . . . 13.
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Notes Indeterminate powers Example Find lim+ (1 − 2x)1/x x→0 Solu on Take 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 . . Notes Indeterminate powers Example Find lim+ (1 − 2x)1/x x→0 Solu on 0 This limit is of the form , so we can use L’Hôpital: 0 −2 ln(1 − 2x) H lim+ = lim+ 1−2x = −2 x→0 x x→0 1 This is not the answer, it’s the log of the answer! So the answer we . want is e−2 . . Notes Another indeterminate power limit Example Find lim (3x)4x x→0 Solu on . . . 14.
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. V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule . Sec on . Notes Summary Form Method 0 0 L’Hôpital’s rule directly ∞ ∞ L’Hôpital’s rule directly ∞ 0·∞ jiggle to make 0 or 0 ∞. ∞ − ∞ combine to make an indeterminate product or quo ent 00 take ln to make an indeterminate product ∞0 di o 1∞ di o . . Notes Final Thoughts L’Hôpital’s Rule only works on indeterminate quo ents Luckily, most indeterminate limits can be transformed into indeterminate quo ents L’Hôpital’s Rule gives wrong answers for non-indeterminate limits! . . Notes . . . 15.
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