Lect11 - Limits at Infinity: Definition: lim f ( x ) = L...

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92.131 Lecture 11 1 of 21 Ronald Brent © 2009 All rights reserved. Limits at Infinity: Definition: L x f x = ) ( lim means that as x grows without bound, the function f gets closer and closer to the value L . L x f x = ) ( lim means that as x decreases without bound, the function f gets closer and closer to the value L . Examples: a) 0 1 lim = x x b) + = = + 2 2 1 1 lim 1 1 1 lim x x x x c) 0 lim = x x e , 0 lim = x x e d) x x sin lim does not exist, since the sine function oscillates forever.

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92.131 Lecture 11 2 of 21 Ronald Brent © 2009 All rights reserved. Algebra of Limits at Infinity Let ±∞ = a , then if L x f a x = ) ( lim and M x g a x = ) ( lim i ) L k x f k a x = ) ( lim , where k is any constant i i ) () M L x g x f a x ± = ± ) ( ) ( lim iii) M L x g x f a x = ) ( ) ( lim i v ) M L x g x f a x = ) ( ) ( lim , provided 0 M
92.131 Lecture 11 3 of 21 Ronald Brent © 2009 All rights reserved. Example: Determine 1 5 2 lim 2 2 + x x x () 5 1 0 5 0 1 1 lim 5 lim 2 lim 1 lim 1 5 2 1 lim 1 1 5 1 2 lim 1 5 2 lim 2 2 2 2 2 2 2 2 2 2 = + = + = + = + = + x x x x x x x x x x x x x x x x x Example: Determine 1 5 2 lim 2 3 + x x x = + = + = + = + = + 0 5 0 1 lim 5 lim 2 lim lim 1 5 2 lim 1 1 5 1 2 lim 1 5 2 lim 2 2 2 2 2 2 2 3 2 3 x x x x x x x x x x x x x x x x x x x

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