University:
Kentucky State University
Course:
MAT 120 | Precalculus
Academic year:

2019
Views:

88

Pages:

4
Author:

Dixie Summers

Limits at Infinity (2) 1. lim(x→∞) x³ = ∞ 2. lim(x→-∞) x³ = -∞ 3. lim(x→∞) 3x³ = ∞ lim(x→-∞) 3x³ = -∞ **Note that:** lim(x→∞) (a₀ + a₁x + a₂x² + ... + aₙxⁿ) = lim(x→∞) aₙxⁿ Examples: 4. lim(x→∞) (3x³ - 9x² + 5) = lim(x→∞) 3x³ = ∞ 5. lim(x→∞) (4x³ + 3x² - 10) = lim(x→∞) 4x³ = ∞ 6. lim(x→∞) (7x³ - 3x² + 1) = lim(x→∞) 7x³ = ∞ 7. lim(x→∞) (3x² - 7x + 1) = lim(x→∞) 3x² = ∞ If the degree of the numerator = the degree of the denominator: 8. lim(x→∞) (7 - 2x + x²) / (3x² + 5) = lim(x→∞) x² / 3x² = lim(x→∞) 1/3 = 1/3 If the degree of the denominator > the degree of the numerator: 9. lim(x→∞) (7 - x²) / (5 + 3x³) = lim(x→∞) -x² / 3x³ = lim(x→∞) -1 / 3x = 0 **Limits at Infinity (Part 2)** Nov 18, 2021 **Note:** * $\sqrt{x^2} = |x|$ * $\sqrt{x^2} = (x^2)^{1/2} = |x|^{1/2} = |x|$ * $\sqrt{x^4} = (x^4)^{1/2} = |x^2| = x^2$ **Example:** lim(x→∞) √(5x^2 + 2) / (x + 3) = √(5x²) / x = lim(x→∞) √5 = √5 or lim(x→∞) √(x²(5 + 2/x²)) / (x(1 + 3/x)) = lim(x→∞) √(5 + 2/x²) / (1 + 3/x) = √5 **Example:** lim(x→∞) √(x^4 + 5) / (x³ + x) = √(x^4) / x³ = x² / x³ = 1 / x = 0 **Example:** lim(x→∞) (-x - 3) / (e^x + 5e^(-x)) = 0 / (∞ + 0) = 0 **Example:** lim(x→∞) (-3e^x) / (5e^x) = -3/5 * e^(x-x) = -3/5 * e^0 = -3/5