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

2020
Views:

331

Pages:

5
Author:

Athena Harrison

L'Hôpital's Rule (Part 1) * Main Idea: For limits of the form 0/0, ∞/∞, ∞ - ∞, 0 · ∞, ∞⁰, 0⁰, 1∞ lim (x/e^x) = ∞/∞ x→∞ L'H → lim (1/e^x) = 1/∞ = 0 x→∞ lim (ln(3x) / ln(2x)) = ∞/∞ x→∞ L'H → lim (3/3x) / (2/2x) = 3/2 x→∞ lim (1 - cos4x) / x² = 0/0 x→0 L'H → lim (4sin4x) / 2x = 4 x→0 lim (5x / e^x) = ∞/∞ x→∞ L'H → lim (5 / e^x) = 5/∞ = 0 x→∞ lim (5^(1/x+x²)) = ∞⁰ x→0 L'H → lim (5(1/x+x²) * ln(5) * (1/x² - 2x)) = 5 * ln(5) * ∞ = ∞ x→0 lim (1 - ln x) / x x→0+ L'H → lim (-1/x) / 1 = -∞ x→0+ lim (x^x) = 1^∞ x→0+ L'H → lim (x^x * (1 + ln x)) = 1 * (1 + (-∞)) = -∞ x→0+ eg) lim ln(sinx) = ∞ x→0+ L'H lim (cosx/sinx) = ∞ x→0+ cosx = secx sinx = tanx → cosx * tanx = cosx * sinx / cosx = sinx lim sinx = 0 x→0+ L'H Rule: 0/0 = ∞/∞ Net L'H Rules: A. Case 1: 0/0 = 0 * ∞ = lim f(x) = 0 = 0 L'H x→a or lim (g(x))^(-1/f(x)) = 1/∞ = 0 x→a B. Case 2: ∞ - ∞ eg) lim x - e^x = ∞ - ∞ x→∞ lim (x - e^x) / (1/x) = ∞/∞ → L'H = lim (1 - e^x) / (-1/x²) = ∞ x→∞ C. Case 3: lim x sin(1/x) = 0 * ∞ x→0 lim sin(1/x) / (1/x) = ∞/∞ → L'H = lim (cos(1/x) * (-1/x²)) / (-1/x²) x→0 lim π cos(1/x) = π x→0 L'Hospital's Rule part 2 **Case (a): 0/0 or ∞/∞** lim (sinx/x) = 0/0 x→0+ L'H → lim (cosx/1) = 1 x→0+ lim (x - sinx) / x³ = 0/0 x→0+ L'H → lim (1 - cosx) / 3x² = 0/0 x→0+ L'H → lim (sinx) / 6x = 0/0 x→0+ L'H → lim (cosx) / 6 = 1/6 x→0+ lim (x - e^x) / x² = ∞/∞ x→∞ L'H → lim (1 - e^x) / 2x = ∞/∞ x→∞ L'H → lim (-e^x) / 2 = -∞/2 = -∞ x→∞ lim (x/e^x) = ∞/∞ x→∞ L'H → lim (1/e^x) = 0 x→∞ lim x^(1/x) = ∞⁰ x→∞ L'H → lim (x^(1/x) * ln(x)) = ∞ * 0 x→∞ L'H → lim (x^(1/x) * (1/x + ln(x))) = ∞ * 0 x→∞ L'H → lim (x^(1/x) * (1/x² + 1/x + ln(x))) = 1 + ∞ = ∞ x→∞