Math 103: L Hopital s Rule
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1 Math 103: L Hopital s Rule Ryan Blair University of Pennsylvania Thursday November 3, 2011 Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
2 Outline 1 L Hospital s Rule 2 Review Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
3 Indeterminant forms L Hospital s Rule For some limits evaluation via substation gives meaningless expressions called Indeterminant Forms lim x 0 sin(x) x = 0 0 Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
4 L Hospital s Rule Indeterminant forms For some limits evaluation via substation gives meaningless expressions called Indeterminant Forms lim x 0 sin(x) x = 0 0 lim x x e x = Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
5 L Hospital s Rule Indeterminant forms For some limits evaluation via substation gives meaningless expressions called Indeterminant Forms lim x 0 sin(x) x = 0 0 lim x x e x = lim x π tan(x) 2 1 (x π 2 )2 = Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
6 L Hospital s Rule Indeterminant forms For some limits evaluation via substation gives meaningless expressions called Indeterminant Forms lim x 0 sin(x) x = 0 0 lim x x e x = lim x π tan(x) 2 1 (x π 2 )2 = Other indeterminant forms include 0, 0 0 and 1 Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
7 L Hospital s Rule L Hospital s Rule for 0 0 Theorem Suppose f (a) = g(a) = 0, f and g are differentiable near a and g (x) 0 for x near a but not equal to a, Then if the right-hand limit exists. f (x) lim x a g(x) = lim f (x) x a g (x) Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
8 L Hospital s Rule L Hospital s Rule for 0 0 Theorem Suppose f (a) = g(a) = 0, f and g are differentiable near a and g (x) 0 for x near a but not equal to a, Then if the right-hand limit exists. f (x) lim x a g(x) = lim f (x) x a g (x) The theorem also holds for one-sided limits and infinite limits. Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
9 L Hospital s Rule L Hospital s Rule for 0 0 Theorem Suppose f (a) = g(a) = 0, f and g are differentiable near a and g (x) 0 for x near a but not equal to a, Then if the right-hand limit exists. f (x) lim x a g(x) = lim f (x) x a g (x) The theorem also holds for one-sided limits and infinite limits. This only helps us with indeterminant forms 0 0. Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
10 L Hospital s Rule L Hospital s Rule for Theorem Suppose f (x) and g(x) as x a, then if the right-hand limit exists. f (x) lim x a g(x) = lim f (x) x a g (x) Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
11 L Hospital s Rule L Hospital s Rule for Theorem Suppose f (x) and g(x) as x a, then if the right-hand limit exists. f (x) lim x a g(x) = lim f (x) x a g (x) The theorem also holds for one-sided limits and infinite limits. Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
12 L Hospital s Rule L Hospital s Rule for Theorem Suppose f (x) and g(x) as x a, then if the right-hand limit exists. f (x) lim x a g(x) = lim f (x) x a g (x) The theorem also holds for one-sided limits and infinite limits. Must convert other indeterminant forms to 0 0 or. Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
13 Review Guidelines for Curve Sketching To sketch the graph of y = f (x), Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
14 Review Guidelines for Curve Sketching To sketch the graph of y = f (x), 1 Find the domain of f (x) and any symmetries. Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
15 Review Guidelines for Curve Sketching To sketch the graph of y = f (x), 1 Find the domain of f (x) and any symmetries. 2 Find f (x) and f (x). Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
16 Review Guidelines for Curve Sketching To sketch the graph of y = f (x), 1 Find the domain of f (x) and any symmetries. 2 Find f (x) and f (x). 3 Find the critical points of f and determine the behavior at each. Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
17 Review Guidelines for Curve Sketching To sketch the graph of y = f (x), 1 Find the domain of f (x) and any symmetries. 2 Find f (x) and f (x). 3 Find the critical points of f and determine the behavior at each. 4 Find where the graph of f is increasing and decreasing. Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
18 Review Guidelines for Curve Sketching To sketch the graph of y = f (x), 1 Find the domain of f (x) and any symmetries. 2 Find f (x) and f (x). 3 Find the critical points of f and determine the behavior at each. 4 Find where the graph of f is increasing and decreasing. 5 Find the points of inflection and the concavity of f. Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
19 Review Guidelines for Curve Sketching To sketch the graph of y = f (x), 1 Find the domain of f (x) and any symmetries. 2 Find f (x) and f (x). 3 Find the critical points of f and determine the behavior at each. 4 Find where the graph of f is increasing and decreasing. 5 Find the points of inflection and the concavity of f. 6 Identify any asymptotes. Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
20 Review Guidelines for Curve Sketching To sketch the graph of y = f (x), 1 Find the domain of f (x) and any symmetries. 2 Find f (x) and f (x). 3 Find the critical points of f and determine the behavior at each. 4 Find where the graph of f is increasing and decreasing. 5 Find the points of inflection and the concavity of f. 6 Identify any asymptotes. 7 Plot key points and asymptotes, and sketch the curve. Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, / 6
Math 103 Day 13: The Mean Value Theorem and Tuesday How Derivatives OctoberShape 26, 2010 a Graph1 / 12
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