Limits and Derivatives

Chapter 4 Applications of Differentiation4.4 Indeterminate Forms and LHpitals RuleWe want to be able to find the limit of the following functions:

However, the methods we learned in Chapter 2 do not work. Because of this, problem a is called an indeterminate form of type 0/0 and problem b is called an indeterminate form of /. We need to use another method.

4.4 Indeterminate Forms and LHpitals Rule2Applications of Calculus

Dr. Erickson

4.4 Indeterminate Forms and LHpitals Rule3Calculus: the Musical! by MatheatreDr. Erickson

3Suppose f and g are differentiable and g(x)0 on an open interval I that contains a (except possibly at a). Suppose that

(In other words, we have an indeterminate form) Then,

If the limit on the right side exists or is or -.4.4 Indeterminate Forms and LHpitals Rule4LHopitals Rule

Dr. EricksonNote 1:LHpitals Rule says that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives, provided that the given conditions are satisfied. It is especially important to verify the conditions regarding the limits of f and g before using lHpitals Rule.

Note 2:LHpitals Rule is also valid for one-sided limits and for limits at infinity or negative infinity; that is, x a can be replaced by any of the symbols x a+, x a, x , or x .

Note 3:Sometimes LHpital is spelled LHospital. Just be aware that the s is silent!

4.4 Indeterminate Forms and LHpitals Rule5NotesDr. EricksonFind4.4 Indeterminate Forms and LHpitals Rule6Example 1

Dr. Erickson4.4 Indeterminate Forms and LHpitals Rule7Indeterminate Products

Dr. EricksonFind the limit. Use lHpitals Rule where appropriate.

4.4 Indeterminate Forms and LHpitals Rule8Example 2

Dr. Erickson4.4 Indeterminate Forms and LHpitals Rule9Indeterminate DifferencesDr. EricksonFind the limit. Use lHpitals Rule where appropriate.

4.4 Indeterminate Forms and LHpitals Rule10Example 3

Dr. EricksonSeveral indeterminate forms arise from the limit: Options:1. and Type: 00

2. and Type: 0

3. and Type: 14.4 Indeterminate Forms and LHpitals Rule11Indeterminate Powers

Dr. EricksonIn each of the three cases, we need to take the natural logarithm of both sides.

NOTE: 0 is NOT indeterminate!4.4 Indeterminate Forms and LHpitals Rule12Indeterminate Powers (continued)Dr. EricksonFind the limit. Use lHpitals Rule where appropriate.

4.4 Indeterminate Forms and LHpitals Rule13Example 4

Dr. EricksonFind the limit. Use lHpitals Rule where appropriate.

4.4 Indeterminate Forms and LHpitals Rule14Example 5

Dr. EricksonL'hopitalicanhasmathCalculus: The Musical! 3rd Edition., track 172013142216.67eng - Visit http://icanhasmath.bandcamp.com