Limits at Infinity and Infinite Limits

Home , Infinity, Limit (mathematics)

Limits at Inﬁnity and Inﬁnite Limits

more examples of limits

– Typeset by FoilTEX – 1 Motivation: handling inﬁnite variable and inﬁnite function

– Typeset by FoilTEX – 2 2

Question. Can we describe f(x)=1/x in mathematics: 1

(1) inﬁnite value of variable? O (2) inﬁnite value of function? 2

Question. Can we describe f(x)=1/x in mathematics: 1

(1) inﬁnite value of variable? O (2) inﬁnite value of function?

Application: horizontal and vertical asymptotes

– Typeset by FoilTEX – 3 Limits at inﬁnity inﬁnite value of variable

– Typeset by FoilTEX – 4 Deﬁnition. L = lim f(x) x→∞ ⇔ ∀ε > 0 ∃N > 0 / x > N ⇒ |f(x) − L| < ε Deﬁnition. L = lim f(x) x→∞ ⇔ ∀ε > 0 ∃N > 0 / x > N ⇒ |f(x) − L| < ε

5x + 1 Example: lim = 5 x→∞ x − 2 ⇔ ∀ε > 0 ∃N > 0 / x > N

5x + 1 ⇒ − 5 < ε x − 2 5x+1 11 11 (in particular, | x−2 − 5| = |x−2| < ε if N = 2 + ε )

– Typeset by FoilTEX – 5 Properties: Let lim f(x) and lim g(x) exist, x→∞ x→∞ lim [f(x) + g(x)] = lim f(x) + lim g(x) x→∞ x→∞ x→∞ lim [f(x) − g(x)] = lim f(x) − lim g(x) x→∞ x→∞ x→∞ lim [f(x) · g(x)] = [ lim f(x)] · [ lim g(x)] x→∞ x→∞ x→∞ lim [f(x)/g(x)] = [ lim f(x)]/[ lim g(x)], x→∞ x→∞ x→∞ lim g(x) 6= 0 x→∞ pn q lim g(x) = n lim g(x), lim g(x) > 0, n even x→∞ x→∞ x→∞

– Typeset by FoilTEX – 6 EXAMPLES

– Typeset by FoilTEX – 7 EXAMPLE 1. Evaluate limit 1 lim x→∞ x EXAMPLE 1. Evaluate limit 1 lim x→∞ x As variable x gets larger, 1/x gets smaller because 1 is being divided by a laaaaaaaarge number: 1 1 x = 1010, = x 1010 EXAMPLE 1. Evaluate limit 1 lim x→∞ x As variable x gets larger, 1/x gets smaller because 1 is being divided by a laaaaaaaarge number: 1 1 x = 1010, = x 1010 The limit is 0. 1 lim = 0. x→∞ x

– Typeset by FoilTEX – 8 EXAMPLE 2. Limits 1 lim = 0 x→∞ xn 1 lim √ = 0 x→∞ n x EXAMPLE 2. Limits 1 lim = 0 x→∞ xn 1 lim √ = 0 x→∞ n x By the same argument 1 1 lim = 0, lim √ = 0 x→∞ x − 100 x→∞ n x − 10000

– Typeset by FoilTEX – 9 EXAMPLE 3. Evaluate limit 1 lim √ x→∞ n x2 + x − C EXAMPLE 3. Evaluate limit 1 lim √ x→∞ n x2 + x − C Recall the graph of

y = x2 + x − C

Observation: y= x 2 + x - C

x → ∞ ⇒ y → ∞

O Value of C does not matter. -1/4-C The answer is 0.

– Typeset by FoilTEX – 10 EXAMPLE 4. Evaluate limit (not evoke graphs) 1 lim x→∞ x2 − x − 1 EXAMPLE 4. Evaluate limit (not evoke graphs) 1 lim x→∞ x2 − x − 1 Attention: indeterminacy ∞ − ∞ 1 = lim x→∞ 1 1 x2 1 − − x x2 1 1 = lim lim = 0 · 1 = 0 x→∞ x2 x→∞ 1 1 1 − − x x2

– Typeset by FoilTEX – 11 EXAMPLE 5. Evaluate limit 5x3 − 2x2 − 1 lim x→∞ x3 − x + 1 EXAMPLE 5. Evaluate limit 5x3 − 2x2 − 1 lim x→∞ x3 − x + 1 Standard trick: divide by the highest degree of x 5x3 2x2 1 2 1 3 − 3 − 3 5 − − 3 = lim x x x = lim x x x→∞ x3 x 1 x→∞ 1 1 − + 1 − + x3 x3 x3 x2 x3 5 − 0 − 0 = = 5 1 − 0 + 0

– Typeset by FoilTEX – 12 EXAMPLE 6. Evaluate limit x − 2 lim p x→∞ 2x2 − x + 1 EXAMPLE 6. Evaluate limit x − 2 lim p x→∞ 2x2 − x + 1 Standard trick: divide by the highest degree of x x 2 x 2 − − x x x x3 = lim √ = lim r x→∞ 2x2 − x + 1 x→∞ 2x2 x 1 √ − + x2 x2 x2 x2 1 − 0 1 = √ = √ . 2 − 0 + 0 2

– Typeset by FoilTEX – 13 Indeterminacy: unsuitable breaking in limits

– Typeset by FoilTEX – 14 Attention: Indeterminacy [x2 + 1] ∞ [x3 − 1] 0 lim lim x→∞ [x3 − 3] ∞ x→1 [x2 − 1] 0 Wrong breaking! Limit laws do not apply!

lim[1 − cos x][cot x] ( 0 · ∞ ) x→0 Substitution is undeﬁned! p p lim [ x2 + x] − [ x2 − x] ( ∞ − ∞ ) x→∞ Rewrite ﬁrst!

– Typeset by FoilTEX – 15 ∞ Indeterminacy : ∞ [x2 + 1] lim x→∞ [x3 − 3] ∞ Indeterminacy : ∞ [x2 + 1] lim x→∞ [x3 − 3] division by highest exponent of x: x2 1 + 3 3 0 + 0 lim x x = = 0 x→∞ x3 3 1 − 0 − x3 x3

– Typeset by FoilTEX – 16 0 Indeterminacy : 0 [x3 − 1] lim x→1 [x2 − 1] 0 Indeterminacy : 0 [x3 − 1] lim x→1 [x2 − 1] factoring and re-grouping: (x − 1)(x2 + x + 1) (x − 1) (x2 + x + 1) lim = lim · x→1 (x − 1)(x + 1) x→1 (x − 1) (x + 1) (x − 1) (x2 + x + 1) 3 3 = [lim ] · [lim ] = 1 · = x→1 (x − 1) x→1 (x + 1) 2 2

– Typeset by FoilTEX – 17 Indeterminacy 0 · ∞:

lim[1 − cos x] · [cot x] x→0 Indeterminacy 0 · ∞:

lim[1 − cos x] · [cot x] x→0 factoring, re-grouping, and special limits: cos x 1 − cos x x cos x lim(1 − cos x) = lim x→0 sin x x→0 x sin x 1 Indeterminacy 0 · ∞:

lim[1 − cos x] · [cot x] x→0 factoring, re-grouping, and special limits: cos x 1 − cos x x cos x lim(1 − cos x) = lim x→0 sin x x→0 x sin x 1

1 − cos x x cos x = [lim ][lim ][lim ] = 0·1·1 = 0 x→0 x x→0 sin x x→0 1

– Typeset by FoilTEX – 18 Indeterminacy ∞ − ∞: p p lim [ x2 + x] − [ x2 − x] x→∞ Indeterminacy ∞ − ∞: p p lim [ x2 + x] − [ x2 − x] x→∞ multiplying by conjugate, re-grouping: √ √ √ √ ( x2 + x − x2 − x)( x2 + x + x2 − x) = lim √ √ x→∞ 1 ( x2 + x + x2 − x) Indeterminacy ∞ − ∞: p p lim [ x2 + x] − [ x2 − x] x→∞ multiplying by conjugate, re-grouping: √ √ √ √ ( x2 + x − x2 − x)( x2 + x + x2 − x) = lim √ √ x→∞ 1 ( x2 + x + x2 − x) x2 + x − x2 + x = lim √ √ x→∞ x2 + x + x2 − x

– Typeset by FoilTEX – 19 2x = lim √ √ x→∞ x2 + x + x2 − x 2x = lim √ √ x→∞ x2 + x + x2 − x ∞/∞: division by the highest exponent 2x x = lim r r x→∞ x2 x x2 x + + − x2 x2 x2 x2 2 = √ √ = 1 1 + 0 + 1 − 0

– Typeset by FoilTEX – 20 Inﬁnite limits inﬁnite value of function

– Typeset by FoilTEX – 21 Deﬁnition. lim f(x) = ∞ x→c ⇔ ∀M > 0 ∃δ > 0 / 0 < |x − c| < δ ⇒ f(x) > M Deﬁnition. lim f(x) = ∞ x→c ⇔ ∀M > 0 ∃δ > 0 / 0 < |x − c| < δ ⇒ f(x) > M

1 Example: lim = ∞ x→2 (x − 2)2 ⇔ ∀M > 0 ∃δ > 0 / 0 < |x−2| < δ 1 ⇒ > M (x − 2)2 (in particular, 1 > M if δ = √1 ) (x−2)2 M

– Typeset by FoilTEX – 22 Example 7. lim f(x) = −∞ x→c ⇔ ∀M < 0 ∃δ > 0 /

0 < |x − c| < δ ⇒ f(x) < M Example 7. lim f(x) = −∞ x→c ⇔ ∀M < 0 ∃δ > 0 /

0 < |x − c| < δ ⇒ f(x) < M

Example 8. lim f(x) = ∞ x→∞ ⇔ ∀M > 0 ∃N > 0 /

x > N ⇒ f(x) > M

– Typeset by FoilTEX – 23 EXAMPLES

– Typeset by FoilTEX – 24 EXAMPLE 9. Evaluate inﬁnite limit 1 lim x→1− x2 − 1 EXAMPLE 9. Evaluate inﬁnite limit 1 lim x→1− x2 − 1 Factoring and sign analysis: 1 1 = lim = = −∞ x→1− (x − 1)(x + 1) (0−) · (2)

– Typeset by FoilTEX – 25 EXAMPLE 10. Evaluate inﬁnite limit x2 + 3x − 4 lim x→2 x2 − 4x + 4 EXAMPLE 10. Evaluate inﬁnite limit x2 + 3x − 4 lim x→2 x2 − 4x + 4 Factoring and sign analysis: (x + 4)(x − 1) (6) · (1) = lim = = −∞ x→2 (x − 2)2 (0+)

– Typeset by FoilTEX – 26 EXAMPLE 11. Evaluate infinite limit 1 lim x→0 sin x EXAMPLE 11. Evaluate infinite limit 1 lim x→0 sin x Sign analysis for one-sided limits: 1 1 lim = = +∞ x→0+ sin x (0+) EXAMPLE 11. Evaluate infinite limit 1 lim x→0 sin x Sign analysis for one-sided limits: 1 1 lim = = +∞ x→0+ sin x (0+) 1 1 lim = = −∞ x→0− sin x (0−) EXAMPLE 11. Evaluate infinite limit 1 lim x→0 sin x Sign analysis for one-sided limits: 1 1 lim = = +∞ x→0+ sin x (0+) 1 1 lim = = −∞ x→0− sin x (0−) Limit at 0 does not exist

– Typeset by FoilTEX – 27