8 7 Indeterminate Forms LHpitals Rule Objective Recognize

8 -7 Indeterminate Forms & L’Hôpital’s Rule Objective: Recognize limits that produce indeterminate forms, apply L’Hôpital’s Rule Miss Battaglia AP Calculus

The Extended Mean Value Theorem If f and g are differentiable on an open interval (a, b) and continuous on [a, b] such that g’(x)≠ 0 for any x in (a, b), then there exists a point c in (a, b) such that

L’Hôpital’s Rule Let f and g be functions that are differentiable on an open interval (a, b) containing c, except possibly at c itself. Assume that g’(x) ≠ 0 for all x in (a, b), except possibly at c itself. If the limit of f(x)/g(x) as x approaches c produces the indeterminate form 0/0, then Provided the limit on the right exists (or is infinite). This result also applies if the limit of f(x)/g(x) as x approaches c produces any one of the indeterminate forms ∞/∞, (-∞)/∞, ∞/(-∞), or (-∞)/(-∞)

Indeterminate Form 0/0 Evaluate

Indeterminate Form ∞/∞ Evaluate

Applying L’Hôpital’s Rule More Than Once Evaluate

Indeterminate Form 0 ∞ Evaluate

When substitution produces 1∞, 1 -∞, 00, ∞ 0 � Set the limit equal to y � Take the log of both sides � Tweak � L’Hopital’s � Solve for y Rule

Indeterminate Form 00 Evaluate

Classwork Evaluate the limit using techniques from Chapter 1 & 3 then evaluate using L’Hopital’s Rule. 1. Evaluate using L’Hopital’s if necessary 2. 3. 4.

Homework �Read 8. 7 Page 576 #1139 odd, 81, 82