University:
California State University, Northridge
Course:
MATH 150AL | Calculus I Laboratory
Academic year:

2004
Views:

421

Pages:

14
Author:

theewokf6v

Infinite limits infinite value of function 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 =∞ 2 x→2 (x − 2) ⇔ ∀M > 0 ∃δ > 0 / 0 < |x−2| < 1 ⇒ > M (x − 2)2 (in particular, 1 (x−2)2 > M if δ = √1 M ) 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 EXAMPLES EXAMPLE 9. Evaluate infinite limit 1 lim 2 x→1− x − 1 EXAMPLE 9. Evaluate infinite limit 1 lim 2 x→1− x − 1 Factoring and sign analysis: 1 1 = lim = − = −∞ (0 ) · (2) x→1− (x − 1)(x + 1) EXAMPLE 10. Evaluate infinite limit x2 + 3x − 4 lim 2 x→2 x − 4x + 4 EXAMPLE 10. Evaluate infinite limit x2 + 3x − 4 lim 2 x→2 x − 4x + 4 Factoring and sign analysis: (x + 4)(x − 1) (6) · (1) = lim = = −∞ 2 + x→2 (x − 2) (0 ) 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 = + = +∞ (0 ) x→0+ sin x EXAMPLE 11. Evaluate infinite limit 1 lim x→0 sin x Sign analysis for one-sided limits: 1 1 lim = + = +∞ (0 ) x→0+ sin x 1 1 lim = − = −∞ (0 ) x→0− sin x EXAMPLE 11. Evaluate infinite limit 1 lim x→0 sin x Sign analysis for one-sided limits: 1 1 lim = + = +∞ (0 ) x→0+ sin x 1 1 lim = − = −∞ (0 ) x→0− sin x Limit at 0 does not exist

Infinite Limits. Infinite Value of Function

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