L'Hospital's Rule

L'Hospital's Rule - LHospitals Rule Salient points...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
1 L’Hospital’s Rule L’Hospital’s Rule Salient points: Indeterminate forms: h 0/0 h ±∞ / L’Hospital’s Rule Other indeterminate forms: h 0 x (product) h - h Indeterminate powers
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 L’Hospital’s Rule Suppose we are trying to analyze the behavior of the function Although F is not defined when x = 1 , we need to know how F behaves near 1 . In particular, we would like to know the value of the limit: ln ( ) 1 x F x x = - INDETERMINATE FORMS 1 ln lim 1 x x x -
Background image of page 2
3 L’Hospital’s Rule In computing this limit, we can’t apply the laws of limits because the limit of the denominator is 0 . s In fact, although the limit in the expression exists, its value is not obvious because both numerator and denominator approach 0 and is not defined. 0 0 INDETERMINATE FORMS
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
4 L’Hospital’s Rule INDETERMINATE FORM —TYPE 0/0 In general, if we have a limit of the form where both f ( x ) 0 and g ( x ) 0 as x a , then this limit may or may not exist. It is called an indeterminate form of type . s We met some limits of this type in Chapter 2. ( ) lim ( ) x a f x g x 0 0
Background image of page 4
5 L’Hospital’s Rule INDETERMINATE FORMS •For rational functions, we can cancel common factors: •For we have: , which we proved using a geometric argument. x x sin 0 sin lim 1 x x x = 2 1 1 lim ) 1 )( 1 ( ) 1 ( lim 1 lim 1 1 2 2 1 = + = - + - = - - x x x x x x x x x x x x
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
6 L’Hospital’s Rule INDETERMINATE FORMS However, these methods do not work for limits such as the expression: s Hence, in this section, we introduce a systematic method, known as l’Hospital’s Rule, for the evaluation of indeterminate forms. Another situation in which a limit is not obvious occurs when
Background image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/13/2011 for the course MATH 1004 taught by Professor Dewsef during the Spring '11 term at Carleton.

Page1 / 23

L'Hospital's Rule - LHospitals Rule Salient points...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online