Chapter 2(E) - Indeterminate Forms Indeterminate Forms Let f and g be continuous at x = a Suppose f(a = 0 and f(b = 0 Then the limit f x 0 lim is

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Indeterminate Forms
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Indeterminate Forms ( ) 0 lim is of the form ( ) 0 xa fx gx Let and be continuous at . f g = Suppose ( ) 0 and ( ) 0. f a fb == Then the limit ( ) 0 because ( ) 0 fa ga =
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0 Indeterminate form 0 ( ) 0 lim ( ) 0 xa fx gx = Use L’Hospital’s Rule
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Who is this person ???
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L’Hospital’s Rule 0 00 (1) and are differentiable in a neighborhood of ; (2) ( ) () 0; (3) ' 0 except possibly at . f gx f x g xx == ( ) ' li m lim ( ) ' x x f x fx g x →→ = Suppose that then
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L’Hospital’s Rule 00 ( ) ' () li m lim ( ) ' x x xx f x fx g x gx →→ = Note: Before applying L' Hospital's Rule ( ) 0 lim is of the form ( ) 0 xa must check that
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L’Hospital’s Rule Suppose ( ) ( ) 0, ' ( ) and ' ( ) exist, and ' ( ) 0. f a g a f a ga == ( ) ' () lim ( ) ' xa f x fa g x = 00 ( ) ' li m lim ( ) ' x x xx f x fx g x gx →→ = In particular,
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0 1 1 1010 (ii) lim 00 x x x + - +- == 1 2 ( 1)1 11 (ii) li m lim () xx d x dx x d x x dx →→    = Can use L'Hospital's rule 1 2 0 1 ( 1) 2 lim 1 x x - + = 1 2 = 0 1 lim 21 x x = + Applying L'H
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0 11 lim x x x +- Question: Can you find without using L'Hospital's rule ???
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0 3 si n 3(0 ) sin 00 (i) lim x xx x -- == ( 3 si n) 3 sin li m lim () d dx d x x dx →→ - - = Can use L'Hospital's rule 0 3 cos lim 1 x x - = 3 cos0 =- 3 1 2 = Applying L'H
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cos0 6 1 6 = = 3 0 si n 0 sin 00 (iii) lim x xx x -- == Can use L'Hospital's rule 32 si n 1 cos li m lim 3 x →→ = Applying L'H 2 1 cos 3(0 )0 - = 0 sin lim 6 x x x = sin 6(0 = 0 cos lim 6 x x =
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0 3 sin lim x xx x - 3 0 sin lim x x - Question: Can you find without using L'Hospital's rule ??? Pause and Think !!!
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22 0 1 co s 1 cos 00 (iv) lim 0 x x xx -- == ++ Can use L'Hospital's rule 2 1 cos sin lim lim 1 2 x →→ - = Applying L'H sin0 1 2(0) = + 0 =
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Question: Can you find without using L'Hospital's rule ???
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This note was uploaded on 11/01/2011 for the course MATH 1505 taught by Professor Yap during the Spring '11 term at National University of Singapore.

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Chapter 2(E) - Indeterminate Forms Indeterminate Forms Let f and g be continuous at x = a Suppose f(a = 0 and f(b = 0 Then the limit f x 0 lim is

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