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Lesson 4-4

Lesson 4-4. L’Hospital’s Rule. Quiz. Homework Problem: Max and Min Values 4-1 Find absolute min and max on the given interval x f(x) = ----------- on [0,2] x² + 1 Reading questions: What indeterminate forms does L’Hospital’s Rule apply to?

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Lesson 4-4

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  1. Lesson 4-4 L’Hospital’s Rule

  2. Quiz • Homework Problem: Max and Min Values 4-1 Find absolute min and max on the given interval x f(x) = ----------- on [0,2] x² + 1 • Reading questions: • What indeterminate forms does L’Hospital’s Rule apply to? • What indeterminate forms does L’Hosptial’s Rule does not apply to directly?

  3. Objectives • Use L’Hospital’s Rule to determine limits of an indeterminate form

  4. Vocabulary • Indeterminate Form – (0/0 or ∞/∞) a form that a value cannot be assigned to without more work

  5. L’Hospital’s Rule Suppose f and g are differentiable and g’(x) ≠ 0 near a (except possibly at a). Suppose that lim f(x) = 0 and lim g(x) = 0 or that lim f(x) = ± ∞ and lim g(x) = ± ∞ (We have an indeterminate form for lim f(x)/g(x)) Then f(x) f’(x) lim --------- = lim -------- g(x) g’(x) if the limit on the right side exists (or is ±∞) x  a x  a x  a x  a x  a x  a

  6. Example of L’Hospital’s Rule sin x lim ------------ = 1 x x→0 Remember that we used the “Squeeze” Theorem in Chapter 2 to show that the follow limit exists: Now using L’Hospital’s Rule we can prove it a different way: Let f(x) = sin x g(x) = x f’(x) = cos x g’(x) = 1 sin x 0 f’(x) cos x 1 lim ------------ = -----  lim -------- = lim ---------- = ----- = 1 x 0 g’(x) 1 1 x→0 x→0 x→0

  7. Indeterminate Forms • The following indeterminate forms we can apply L’Hospital’s Rule to directly • These forms we have to use logarithmic manipulation on before and after • These forms we have to convert into a quotient to apply L’Hospital’s Rule to 0 ∞ --- and ----- 0 ∞ 00 1∞ ∞0 0 ∙ ∞ or ∞ - ∞

  8. Example 1 ln x Find lim ------------- x→1 x - 1 1/x = lim --------- = 1 1 x→1 Check to see if L’Hospital’s Rule applies: lim ln x = 0 lim x-1 = 0 Yes! x→1 x→1 f(x) = ln x so f’(x) = 1/x g(x) = x – 1 so g’(x) = 1

  9. Example 1 ln x Find lim ------------- x→1 x - 1 1/x = lim --------- = 1 1 x→1 Check to see if L’Hospital’s Rule applies: lim ln x = 0 lim x-1 = 0 Yes! x→1 x→1 f(x) = ln x so f’(x) = 1/x g(x) = x – 1 so g’(x) = 1

  10. Example 2 ex lim --------- = ∞ 2 ex Find lim ---------- x→∞ x² ex = lim --------- = 2x x→∞ x→∞ Check to see if L’Hospital’s Rule applies: lim ex = ∞ lim x² = ∞ Yes! x→∞ x→∞ f(x) = ex so f’(x) = ex g(x) = x² so g’(x) = 2x f(x) = ex so f’(x) = ex g(x) = 2x so g’(x) = 2

  11. Example 3 sin x Find lim -------------- x→π- 1 – cos x Check to see if L’Hospital’s Rule applies: lim sin x = 0 lim 1 - cos x = 2 No! x→π x→π

  12. Summary & Homework • Summary: • L’Hospital’s Rule applies to indeterminate quotients in the form of 0/0 or ∞/∞ • Other indeterminate forms exist and can be solved for, but are beyond the scope of this course • Homework: • pg 313-315: 1, 5, 7, 12, 15, 17, 27, 28, 40, 49, 53, 55

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