Like this presentation? Why not share!

# Lesson 6: Limits Involving Infinity

## by Matthew Leingang, Clinical Associate Professor of Mathematics at New York University on Feb 05, 2010

• 2,623 views

Infinity is a tricky thing. It's tempting to treat it as a special number, but that can lead to trouble. In this slideshow we look at the different kinds of infinite limits and limits at infinity.

Infinity is a tricky thing. It's tempting to treat it as a special number, but that can lead to trouble. In this slideshow we look at the different kinds of infinite limits and limits at infinity.

### Views

Total Views
2,623
Views on SlideShare
2,607
Embed Views
16

Likes
0
69
2

### 2 Embeds16

 http://www.slideshare.net 11 http://a0.twimg.com 5

### Report content

12 of 2 previous next

## Lesson 6: Limits Involving InfinityPresentation Transcript

• Section 1.6 Limits involving Inﬁnity V63.0121.006/016, Calculus I February 3, 2010 Announcements Ofﬁce Hours: M,W 1:30–2:30, R 9–10 (CIWW 726) Written Assignment #2 due today. WebAssignments due Tuesday. First Quiz: Friday February 12 in recitation (§§1.1–1.4) . . . . . .
• 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. . . . . . .
• 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. . . . . . .
• 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. . . . . . .
• 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. . . . . . .
• 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. . . . . . .
• 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 . . . . . .
• 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 . . . . . . .
• 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. “Large” takes the place . x . of “close to L”. . . . . . .
• 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. “Large” takes the place . x . of “close to L”. . . . . . .
• 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. “Large” takes the place . x . of “close to L”. . . . . . .
• 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. “Large” takes the place . x . of “close to L”. . . . . . .
• 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. “Large” takes the place . x . of “close to L”. . . . . . .
• 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. “Large” takes the place . x . of “close to L”. . . . . . .
• 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. “Large” takes the place . x . of “close to L”. . . . . . .
• 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. . . . . . .
• 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. . . . . . .
• 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 − . . . . . .
• Inﬁnite Limits we Know y . . . 1 lim + x =∞ x →0 . . . . . . . . x . . . . . . . . . .
• Inﬁnite Limits we Know y . . . 1 lim + x =∞ x →0 . 1 lim = −∞ x →0 − x . . . . . . . x . . . . . . . . . .
• Inﬁnite Limits we Know y . . . 1 lim + x =∞ x →0 . 1 lim = −∞ x →0 − x . . . . . . . x . 1 lim =∞ x →0 x 2 . . . . . . . . .
• 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 x →a − x→a+ continuous. . . . . . .
• 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 x →a − x→a+ continuous. Solution The denominator factors as (x − 1)(x − 2). We can record the signs of the factors on the number line. . . . . . .
• Use the number line − .. 0 .. . + . x − 1) ( 1 . So . . . . . .
• Use the number line − .. 0 .. . + . x − 1) ( 1 . − . 0 .. . + . x − 2) ( 2 . So . . . . . .
• Use the number line − .. 0 .. . + . x − 1) ( 1 . − . 0 .. . + . x − 2) ( 2 . . + . x2 + 2 ) ( So . . . . . .
• Use the number line − .. 0 .. . + . x − 1) ( 1 . − . 0 .. . + . x − 2) ( 2 . . + . x2 + 2 ) ( . . f .(x) 1 . 2 . So . . . . . .
• Use the number line − .. 0 .. . + . x − 1) ( 1 . − . 0 .. . + . x − 2) ( 2 . . + . x2 + 2 ) ( . + . . f .(x) 1 . 2 . So . . . . . .
• Use the number line − .. 0 .. . + . x − 1) ( 1 . − . 0 .. . + . x − 2) ( 2 . . + . x2 + 2 ) ( . + . ∞. + . f .(x) 1 . 2 . So lim f(x) = +∞ x →1 − . . . . . .
• Use the number line − .. 0 .. . + . x − 1) ( 1 . − . 0 .. . + . x − 2) ( 2 . . + . x2 + 2 ) ( . + . ∞ .. ∞ + − . f .(x) 1 . 2 . So lim f(x) = +∞ x →1 − lim f(x) = −∞ x →1 + . . . . . .
• Use the number line − .. 0 .. . + . x − 1) ( 1 . − . 0 .. . + . x − 2) ( 2 . . + . x2 + 2 ) ( . + . ∞ .. ∞ . + − − . f .(x) 1 . 2 . So lim f(x) = +∞ x →1 − lim f(x) = −∞ x →1 + . . . . . .
• Use the number line − .. 0 .. . + . x − 1) ( 1 . − . 0 .. . + . x − 2) ( 2 . . + . x2 + 2 ) ( . + . ∞ .. ∞ . . ∞ . + − − − f .(x) 1 . 2 . So lim f(x) = +∞ lim f(x) = −∞ x →1 − x →2 − lim f(x) = −∞ x →1 + . . . . . .
• Use the number line − .. 0 .. . + . x − 1) ( 1 . − . 0 .. . + . x − 2) ( 2 . . + . x2 + 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 + . . . . . .
• Use the number line − .. 0 .. . + . x − 1) ( 1 . − . 0 .. . + . x − 2) ( 2 . . + . x2 + 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 + . . . . . .
• Use the number line − .. 0 .. . + . x − 1) ( 1 . − . 0 .. . + . x − 2) ( 2 . . + . x2 + 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 + . . . . . .
• 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 +∞.” . . . . . .
• The graph so far y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . .
• The graph so far y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . .
• The graph so far y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . .
• The graph so far y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . .
• The graph so far y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . .
• Limit Laws (?) with inﬁnite limits If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞. x →a x→a x →a That is, ∞ . . +∞=∞ If lim f(x) = −∞ and lim g(x) = −∞, then x →a x →a lim (f(x) + g(x)) = −∞. That is, x →a − . . ∞ − ∞ = −∞ . . . . . .
• Rules of Thumb with inﬁnite limits If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞. x →a x→a x →a That is, ∞ . . +∞=∞ If lim f(x) = −∞ and lim g(x) = −∞, then x →a x →a lim (f(x) + g(x)) = −∞. That is, x →a − . . ∞ − ∞ = −∞ . . . . . .
• Rules of Thumb with inﬁnite limits If lim f(x) = L and lim g(x) = ±∞, then x →a x →a lim (f(x) + g(x)) = ±∞. That is, x →a L+∞=∞ . . L − ∞ = −∞ . . . . . .
• Rules of Thumb with inﬁnite limits Kids, don’t try this at home! 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. { −∞ if L > 0 . . · (−∞) = L ∞ if L < 0. . . . . . .
• Multiplying inﬁnite limits Kids, don’t try this at home! The product of two inﬁnite limits is inﬁnite. ∞·∞=∞ . . ∞ · (−∞) = −∞ (−∞) · (−∞) = ∞ . . . . . .
• Dividing by Inﬁnity Kids, don’t try this at home! The quotient of a ﬁnite limit by an inﬁnite limit is zero: L . . =0 ∞ . . . . . .
• 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. . . . . . .
• Indeterminate Limit forms L Limits of the form are indeterminate. There is no rule for 0 evaluating such a form; the limit must be examined more closely. Consider these: 1 −1 lim =∞ lim = −∞ x→0 x2 x →0 x 2 1 1 lim + x =∞ lim = −∞ x →0 x →0 − x 1 L Worst, lim is of the form , but the limit does not x→0 x sin(1/x) 0 exist, even in the left- or right-hand sense. There are inﬁnitely many vertical asymptotes arbitrarily close to 0! . . . . . .
• 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 x →0 + x 1. 1 The limit lim sin2 x · is of the form 0 · ∞, but the answer is x →0 + x 0. 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. . . . . . .
• Indeterminate forms are like Tug Of War Which side wins depends on which side is stronger. . . . . . .
• 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 . . . . . .
• 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 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→−∞ . . . . . .
• 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! . . . . . .
• Basic limits at inﬁnity Theorem Let n be a positive integer. Then 1 lim =0 x→∞ xn 1 lim n = 0 x→−∞ x . . . . . .
• Using the limit laws to compute limits at ∞ Example Find 2x3 + 3x + 1 lim x→∞ 4x3 + 5x2 + 7 if it exists. A does not exist B 1/2 C 0 D ∞ . . . . . .
• Using the limit laws to compute limits at ∞ Example Find 2x3 + 3x + 1 lim x→∞ 4x3 + 5x2 + 7 if it exists. A does not exist B 1/2 C 0 D ∞ . . . . . .
• 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 . . . . . .
• 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. . . . . . .
• Another Example Example x Find lim x→∞ x2 +1 . . . . . .
• Another Example Example x Find lim x→∞ x2 +1 Answer The limit is 0. . . . . . .
• Solution Again, 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 . . . . . .
• Another Example Example x Find lim x→∞ x2 +1 Answer The limit is 0. y . . x . . . . . . .
• Another Example 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. . . . . . .
• Solution Again, 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 ∞. . . . . . .
• Another Example Example Find √ 3x4 + 7 lim x→∞ x2 + 3 . . . . . .
• Another Example Example Find √ √ √ . 3x4 + 7 ∼ . 3x4 = 3x2 √ 3x4 + 7 lim x→∞ x2 + 3 Answer √ The limit is 3. . . . . . .
• Solution √ √ 3x4 + 7 x4 (3 + 7/x4 ) lim = lim 2 x→∞ x2 + 3 x→∞ x (1 + 3/x2 ) √ x2 (3 + 7/x4 ) = lim 2 x→∞ x (1 + 3/x2 ) √ (3 + 7/x4 ) = lim x→∞ 1 + 3/x2 √ 3+0 √ = = 3. 1+0 . . . . . .
• Rationalizing to get a limit Example (√ ) Compute lim 4x2 + 17 − 2x . x→∞ . . . . . .
• 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 4x2 + 17 − 2x = lim 4x2 + 17 − 2x · √ x→∞ x→∞ 4x2 + 17 + 2x (4x 2 + 17) − 4x2 = lim √ x→∞ 4x2 + 17 + 2x 17 = lim √ =0 x→∞ 4x 2 + 17 + 2x . . . . . .
• Kick it up a notch Example (√ ) Compute lim 4x2 + 17x − 2x . x→∞ . . . . . .
• 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→∞ 4x 2 + 17x + 2x x→∞ 4 + 17/x + 2 4 . . . . . .
• 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. . . . . . .