L'Hospital's Rule

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

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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

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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 -
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

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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
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

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

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