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# Lesson 6: Limits Involving Infinity (Section 41 handout)

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• 1. V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010 Notes Section 1.6 Limits involving Inﬁnity V63.0121.041, Calculus I New York University September 21, 2010 Announcements Announcements Notes V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 2 / 35 Objectives Notes “Intuit” limits involving inﬁnity by eyeballing the expression. Show limits involving inﬁnity by algebraic manipulation and conceptual argument. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 3 / 35 1
• 2. V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010 Recall the deﬁnition of limit Notes 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. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 4 / 35 Recall the unboundedness problem Notes 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. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 5 / 35 Outline Notes 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 V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 6 / 35 2
• 3. V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010 Inﬁnite Limits Notes 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. “Large” takes the place of x “close to L”. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 7 / 35 Negative Inﬁnity Notes 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. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 8 / 35 Vertical Asymptotes Notes 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− V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 9 / 35 3
• 4. V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010 Inﬁnite Limits we Know Notes y 1 lim =∞ x→0+ x 1 lim = −∞ x→0− x x 1 lim =∞ x→0 x 2 V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 10 / 35 Finding limits at trouble spots Notes Example Let x2 + 2 f (x) = x 2 − 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. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 11 / 35 Use the number line Notes − 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x 2 + 2) + +∞ −∞ − −∞ +∞ + f (x) 1 2 So lim f (x) = +∞ lim f (x) = −∞ x→1− x→2− lim f (x) = −∞ lim f (x) = +∞ x→1+ x→2+ V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 12 / 35 4
• 5. V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010 In English, now Notes 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 +∞.” V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 13 / 35 The graph so far Notes 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 V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 14 / 35 Rules of Thumb with inﬁnite limits Notes Fact ∞+∞=∞ If lim f (x) = ∞ and x→a lim g (x) = ∞, then x→a −∞ − ∞ = ∞ lim (f (x) + g (x)) = ∞. x→a If lim f (x) = −∞ and x→a lim g (x) = −∞, then x→a lim (f (x) + g (x)) = −∞. x→a V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 15 / 35 5
• 6. V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010 Rules of Thumb with inﬁnite limits Notes L+∞=∞ L − ∞ = −∞ Fact If lim f (x) = L and lim g (x) = ±∞, then x→a x→a lim (f (x) + g (x)) = ±∞. x→a V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 16 / 35 Rules of Thumb with inﬁnite limits Kids, don’t try this at home! Notes ∞ 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 L > 0 L · (−∞) = ∞ if L < 0. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 17 / 35 Multiplying inﬁnite limits Kids, don’t try this at home! Notes ∞·∞=∞ ∞ · (−∞) = −∞ (−∞) · (−∞) = ∞ Fact The product of two inﬁnite limits is inﬁnite. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 18 / 35 6
• 7. V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010 Dividing by Inﬁnity Kids, don’t try this at home! Notes L =0 ∞ Fact The quotient of a ﬁnite limit by an inﬁnite limit is zero. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 19 / 35 Dividing by zero is still not allowed Notes 1 =∞ 0 There are examples of such limit forms where the limit is ∞, −∞, undecided between the two, or truly neither. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 20 / 35 Indeterminate Limit forms Notes 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→0x sin(1/x) 0 in the left- or right-hand sense. There are inﬁnitely many vertical asymptotes arbitrarily close to 0! V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 21 / 35 7
• 8. V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010 Indeterminate Limit forms Notes 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. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 22 / 35 Indeterminate forms are like Tug Of War Notes Which side wins depends on which side is stronger. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 23 / 35 Outline Notes 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 V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 24 / 35 8
• 9. V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010 Deﬁnition Notes 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! V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 25 / 35 Basic limits at inﬁnity Notes Theorem Let n be a positive integer. Then 1 lim =0 x→∞ x n 1 lim =0 x→−∞ x n V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 26 / 35 Using the limit laws to compute limits at ∞ Notes 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 heuristic deﬁnitions of asymptote. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 27 / 35 9
• 10. V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010 Solution Notes 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/x 2 ) x 1 + 1/x 2 x 1 1 1 1 lim = lim = lim · lim x→∞ x 2 + 1 x→∞ x 1 + 1/x 2 x→∞ x x→∞ 1 + 1/x 2 1 =0· = 0. 1+0 Remark Had the higher power been in the numerator, the limit would have been ∞. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 28 / 35 Another Example Notes 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 ∞ V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 29 / 35 Solution Notes V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 30 / 35 10
• 11. V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010 Still Another Example Notes Example Find √ 3x 4 + 7 lim x→∞ x2 + 3 V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 31 / 35 Solution Notes V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 32 / 35 Rationalizing to get a limit Notes 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. √ 4x 2 + 17 + 2x lim 4x 2 + 17 − 2x = lim 4x 2 + 17 − 2x · √ x→∞ x→∞ 4x 2 + 17 + 2x (4x 2 + 17) − 4x 2 = lim √ x→∞ 4x 2 + 17 + 2x 17 = lim √ =0 x→∞ 4x 2 + 17 + 2x V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 33 / 35 11
• 12. V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010 Kick it up a notch Notes Example Compute lim 4x 2 + 17x − 2x . x→∞ V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 34 / 35 Summary Notes 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. V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 35 / 35 Notes 12