Lesson 4-4 L’Hospital’s Rule

Quiz Homework Problem: Max and Min Values 4-1 x² + 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?

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

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

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

Example of L’Hospital’s Rule Remember that we used the “Squeeze” Theorem in Chapter 2 to show that the follow limit exists: sin x lim ------------ = 1 x x→0 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

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 ----- 00 1∞ ∞0 0 ∙ ∞ or ∞ - ∞

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

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

Example 2 ex Find lim ---------- x→∞ x² ex = lim --------- = 2x ex Check to see if L’Hospital’s Rule applies: lim ex = ∞ lim x² = ∞ Yes! 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

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

Summary & Homework Summary: Homework: 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