5.8 Indeterminate forms and L Hôpital s rule

Transcription

1 5.8 Indeterminate forms and L Hôpital s rule Mark Woodard Furman U Fall 2009 Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

2 Outline 1 The forms 0/0 and / 2 Examples 3 Indeterminate products: 0 4 Indeterminate differences: 5 Indeterminate powers: 0 0, 0 and 1 Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

3 The forms 0/0 and / Definition We say that the limit is of the form 0/0 or / if f (x) lim x a g(x) lim f (x) = lim g(x) = 0 or lim f (x) = lim g(x) = ±, x a x a x a x a respectively. These are called indeterminate. In this context, a can be any of a +, a, ±. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

4 The forms 0/0 and / Theorem (L Hôpital s Rule) If lim x a f (x)/g(x) is an ideterminate form of type 0/0 or / and if f (x) lim x a g (x) = L, then f (x) lim x a g(x) = L. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

5 The forms 0/0 and / Theorem (L Hôpital s Rule) If lim x a f (x)/g(x) is an ideterminate form of type 0/0 or / and if f (x) lim x a g (x) = L, then f (x) lim x a g(x) = L. Remark When facing an indeterminate form, students will often write: f (x) lim x a g(x) = lim f (x) x a g (x) which is, strictly speaking, wrong. L Hôpital s rule states that these are equal only when the limit on the right exists. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

6 Examples Evaluate the following limits: Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

7 Examples Evaluate the following limits: lim x 0 sin(x) x Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

8 Examples Evaluate the following limits: lim x 0 sin(x) x lim x 1 x 2 1 x 1 Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

9 Examples Evaluate the following limits: lim x 0 sin(x) x lim x 1 x 2 1 x 1 lim x x 4 e x. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

10 Indeterminate products: 0 Example Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

11 Indeterminate products: 0 Example Evaluate the limit: lim x 0 + x ln(x). Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

12 Indeterminate products: 0 Example Evaluate the limit: lim x 0 + x ln(x). You can see the difficulty. The first function, x, is trying to take the product to 0; the second function, ln(x), is trying to take the product to. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

13 Indeterminate products: 0 Example Evaluate the limit: lim x 0 + x ln(x). You can see the difficulty. The first function, x, is trying to take the product to 0; the second function, ln(x), is trying to take the product to. This is an indeterminate product of the type 0. How can we resolve this limit? Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

14 Indeterminate products: 0 Example Evaluate the limit: lim x 0 + x ln(x). You can see the difficulty. The first function, x, is trying to take the product to 0; the second function, ln(x), is trying to take the product to. This is an indeterminate product of the type 0. How can we resolve this limit? Strategy for the 0 form Suppose that the limit lim x a f (x)g(x) is of the form 0. This form can be converted into either a 0/0 or an / form by algebra: f (x)g(x) = g(x) 1/f (x) or f (x)g(x) = f (x) 1/g(x). Now the limit can be attacked by the previous methods. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

15 Indeterminate products: 0 Evaluate the following limits: Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

16 Indeterminate products: 0 Evaluate the following limits: Find lim x 0 + x ln(x). Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

17 Indeterminate products: 0 Evaluate the following limits: Find lim x 0 + x ln(x). Find lim x xe x. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

18 Indeterminate products: 0 Evaluate the following limits: Find lim x 0 + x ln(x). Find lim x xe x. Find lim x x ( π/2 tan 1 (x) ). Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

19 Indeterminate differences: The basic strategy for indeterminate differences Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

20 Indeterminate differences: The basic strategy for indeterminate differences An indeterminate difference is any limit of the form ( ) lim f (x) g(x) x a in which f and g simultaneously approach + or. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

21 Indeterminate differences: The basic strategy for indeterminate differences An indeterminate difference is any limit of the form ( ) lim f (x) g(x) x a in which f and g simultaneously approach + or. To handle an form, use algebra to convert this form into one of the other forms. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

22 Indeterminate differences: Find lim u 0 + ( 1 1 e u 1 ). u Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

23 Indeterminate differences: Find lim u 0 + Solution ( 1 1 e u 1 ). u Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

24 Indeterminate differences: Find lim u 0 + ( 1 1 e u 1 ). u Solution This is called an form, since both terms are approaching +. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

25 Indeterminate differences: Find lim u 0 + ( 1 1 e u 1 ). u Solution This is called an form, since both terms are approaching +. We can force a common denominator: 1 1 e u 1 u = u 1 + e u u(1 e u ). Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

26 Indeterminate differences: Find lim u 0 + ( 1 1 e u 1 ). u Solution This is called an form, since both terms are approaching +. We can force a common denominator: 1 1 e u 1 u = u 1 + e u u(1 e u ). As u 0 +, the right-hand-side is now a 0/0 form and can be treated using l Hôpital s rule. The answer is 1/2. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

27 Indeterminate powers: 0 0, 0 and 1 The basic strategy for indeterminate powers Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

28 Indeterminate powers: 0 0, 0 and 1 The basic strategy for indeterminate powers An indeterminate power is any limit of the form lim x a f (x)g(x) resulting in 0 0, 0 and 1. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

29 Indeterminate powers: 0 0, 0 and 1 The basic strategy for indeterminate powers An indeterminate power is any limit of the form lim x a f (x)g(x) resulting in 0 0, 0 and 1. In each of these cases, first write f (x) g(x) = exp ( g(x) ln f (x) ). The exponent, g(x) ln f (x), will be in one the preceding forms and can be handled by those methods. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

30 Indeterminate powers: 0 0, 0 and 1 Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

31 Indeterminate powers: 0 0, 0 and 1 ( Find lim ) x x x 2. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

32 Indeterminate powers: 0 0, 0 and 1 ( Find lim ) x x x 2. Find lim x x 1/x. Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11

33 Indeterminate powers: 0 0, 0 and 1 ( Find lim ) x x x 2. Find lim x x 1/x. Find lim x 0 + x sin(x). Mark Woodard (Furman U) 5.8 Indeterminate forms and L Hôpital s rule Fall / 11