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# Lesson 6: Limits Involving Infinity (handout)

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Infinity is a complicated concept, but there are rules for dealing with both limits at infinity and infinite limits.

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

1. 1. . V63.0121.001: Calculus I . Sec on 1.5: Limits Involving Inﬁnity . February 9, 2011 Notes Sec on 1.5 Limits Involving Inﬁnity V63.0121.001: Calculus I Professor Ma hew Leingang New York University February 9, 2011 . . Notes Announcements Get-to-know-you extra credit due Friday February 11 Quiz 1 is next week in recita on. Covers Sec ons 1.1–1.4 . . Notes Objectives “Intuit” limits involving inﬁnity by eyeballing the expression. Show limits involving inﬁnity by algebraic manipula on and conceptual argument. . . . 1.
2. 2. . V63.0121.001: Calculus I . Sec on 1.5: Limits Involving Inﬁnity . February 9, 2011 Notes Recall the deﬁnition of limit Deﬁni on 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. . . Notes The unboundedness problem y 1 Recall why lim+ doesn’t x→0 x exist. No ma er how thin we draw the strip to the right of x = 0, L? we cannot “capture” the graph inside the box. . x . . Notes Inﬁnite Limits Deﬁni on y The nota on 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 . x to a but not equal to a. “Large” takes the place of “close to L”. . . . 2.
3. 3. . V63.0121.001: Calculus I . Sec on 1.5: Limits Involving Inﬁnity . February 9, 2011 Notes Negative Inﬁnity Deﬁni on The nota on lim f(x) = −∞ x→a means that the values of f(x) can be made arbitrarily large nega ve (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 (nega ve) number. . . Notes Vertical Asymptotes Deﬁni on The line x = a is called a ver cal 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− . . Notes Inﬁnite Limits we Know y 1 lim =∞ x→0+ x 1 lim = −∞ x→0− x . x 1 lim = ∞ x→0 x2 . . . 3.
4. 4. . V63.0121.001: Calculus I . Sec on 1.5: Limits Involving Inﬁnity . February 9, 2011 Notes 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 con nuous. x→a x→a Solu on The denominator factors as (x − 1)(x − 2). We can record the signs of the factors on the number line. . . Notes Use the number line small small −. 0 + (x − 1) small 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+ . . Notes In English, now To explain the limit, you can say: “As x → 1− , the numerator approaches 3, and the denominator approaches 0 while remaining posi ve. So the limit is +∞.” . . . 4.
5. 5. . V63.0121.001: Calculus I . Sec on 1.5: Limits Involving Inﬁnity . February 9, 2011 Notes 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 . . Notes Rules of Thumb with inﬁnite limits Fact The sum of two posi ve or two nega ve 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 x→a x→a lim (f(x) + g(x)) = −∞. x→a Remark We don’t say anything here about limits of the form ∞ − ∞. . . Notes Rules of Thumb with inﬁnite limits Fact The sum of a ﬁnite limit and an inﬁnite limit is inﬁnite. If lim f(x) = L and lim g(x) = ±∞, then x→a x→a lim (f(x) + g(x)) = ±∞. x→a . . . 5.
6. 6. . V63.0121.001: Calculus I . Sec on 1.5: Limits Involving Inﬁnity . February 9, 2011 Notes Rules of Thumb with inﬁnite limits 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 x→a x→a lim f(x) · g(x) = ∞. 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 x→a x→a lim f(x) · g(x) = −∞. 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 . Notes Multiplying inﬁnite limits 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 . . Notes Dividing by Inﬁnity Fact The quo ent 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) . . . 6.
7. 7. . V63.0121.001: Calculus I . Sec on 1.5: Limits Involving Inﬁnity . February 9, 2011 Notes 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. . . Notes Indeterminate Limit forms L Limits of the form are indeterminate. There is no rule for 0 evalua ng 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, x→0 x sin(1/x) 0 even in the le - or right-hand sense. There are inﬁnitely many . ver cal asymptotes arbitrarily close to 0! . Notes 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. . . . 7.
8. 8. . V63.0121.001: Calculus I . Sec on 1.5: Limits Involving Inﬁnity . February 9, 2011 Notes Indeterminate forms are like Tug Of War Which side wins depends on which side is stronger. . . Notes Outline Ver cal Asymptotes Inﬁnite Limits we Know Limit “Laws” with Inﬁnite Limits Indeterminate Limit forms Limits at ∞ Algebraic rates of growth Ra onalizing to get a limit . . Notes Limits at Inﬁnity Deﬁni on Let f be a func on 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. . . . 8.
9. 9. . V63.0121.001: Calculus I . Sec on 1.5: Limits Involving Inﬁnity . February 9, 2011 Notes Horizontal Asymptotes Deﬁni on 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! . . Notes Basic limits at inﬁnity Theorem Let n be a posi ve integer. Then 1 lim =0 x→∞ xn 1 lim =0 x→−∞ xn . . Notes Limit laws at inﬁnity Fact Any limit law that concerns ﬁnite limits at a ﬁnite point a is s ll 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 . . 9.
10. 10. . V63.0121.001: Calculus I . Sec on 1.5: Limits Involving Inﬁnity . February 9, 2011 Using the limit laws to compute Notes limits at ∞ Example x Find lim x→∞ x2 +1 Answer The limit is 0. No ce that the graph does cross the y asymptote, which contradicts one of the . x commonly held beliefs of what an asymptote is. . . Notes Solution Solu on . . Notes 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 ∞ . . . 10.
11. 11. . V63.0121.001: Calculus I . Sec on 1.5: Limits Involving Inﬁnity . February 9, 2011 Notes Solution Solu on . . Notes Still Another Example Example Find √ 3x4 + 7 lim x→∞ x2 + 3 . . Notes Solution Solu on . . . 11.
12. 12. . V63.0121.001: Calculus I . Sec on 1.5: Limits Involving Inﬁnity . February 9, 2011 Notes Rationalizing to get a limit . Example (√ ) Compute lim 4x2 + 17 − 2x . x→∞ Solu on This limit is of the form ∞ − ∞, which we cannot use. So we ra onalize the numerator (the denominator is 1) to get an expression that we can use the limit laws on. (√ ) (√ ) √4x2 + 17 + 2x lim 4x2 + 17 − 2x = lim 4x2 + 17 − 2x · √ x→∞ x→∞ 4x2 + 17 + 2x (4x2 + 17) − 4x2 17 = lim √ = lim √ =0 x→∞ 4x2 + 17 + 2x x→∞ 4x2 + 17 + 2x . . Notes Kick it up a notch Example (√ ) Compute lim 4x2 + 17x − 2x . x→∞ Solu on . . Notes Summary Inﬁnity is a more complicated concept than a single number. There are rules of thumb, but there are also excep ons. 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. . . . 12.