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Calculus_Cheat_Sheet - Calculus Cheat Sheet Calculus Cheat...

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Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Limits Definitions Precise Definition : We say ( ) lim x a f x L fi = if for every 0 e > there is a 0 d > such that whenever 0 x a d < - < then ( ) f x L e - < . “Working” Definition : We say ( ) lim x a f x L fi = if we can make ( ) f x as close to L as we want by taking x sufficiently close to a (on either side of a ) without letting x a = . Right hand limit : ( ) lim x a f x L + fi = . This has the same definition as the limit except it requires x a > . Left hand limit : ( ) lim x a f x L - fi = . This has the same definition as the limit except it requires x a < . Limit at Infinity : We say ( ) lim x f x L fi¥ = if we can make ( ) f x as close to L as we want by taking x large enough and positive. There is a similar definition for ( ) lim x f x L fi-¥ = except we require x large and negative. Infinite Limit : We say ( ) lim x a f x fi = ¥ if we can make ( ) f x arbitrarily large (and positive) by taking x sufficiently close to a (on either side of a ) without letting x a = . There is a similar definition for ( ) lim x a f x fi = -¥ except we make ( ) f x arbitrarily large and negative. Relationship between the limit and one-sided limits ( ) lim x a f x L fi = ° ( ) ( ) lim lim x a x a f x f x L + - fi fi = = ( ) ( ) lim lim x a x a f x f x L + - fi fi = = ° ( ) lim x a f x L fi = ( ) ( ) lim lim x a x a f x f x + - fi fi ° ( ) lim x a f x fi Does Not Exist Properties Assume ( ) lim x a f x fi and ( ) lim x a g x fi both exist and c is any number then, 1. ( ) ( ) lim lim x a x a cf x c f x fi fi = Ø ø º û 2. ( ) ( ) ( ) ( ) lim lim lim x a x a x a f x g x f x g x fi fi fi = Ø ø º û 3. ( ) ( ) ( ) ( ) lim lim lim x a x a x a f x g x f x g x fi fi fi = Ø ø º û 4. ( ) ( ) ( ) ( ) lim lim lim x a x a x a f x f x g x g x fi fi fi Ø ø = OE oe º û provided ( ) lim 0 x a g x fi 5. ( ) ( ) lim lim n n x a x a f x f x fi fi Ø ø = Ø ø º û º û 6. ( ) ( ) lim lim n n x a x a f x f x fi fi Ø ø = º û Basic Limit Evaluations at – ¥ Note : ( ) sgn 1 a = if 0 a > and ( ) sgn 1 a = - if 0 a < . 1. lim x x fi¥ = ¥ e & lim 0 x x fi- ¥ = e 2. ( ) lim ln x x fi¥ = ¥ & ( ) 0 lim ln x x - fi = -¥ 3. If 0 r > then lim 0 r x b x fi¥ = 4. If 0 r > and r x is real for negative x then lim 0 r x b x fi-¥ = 5. n even : lim n x x fi–¥ = ¥ 6. n odd : lim n x x fi¥ = ¥ & lim n x x fi- ¥ = -¥ 7. n even : ( ) lim sgn n x a x b x c a fi–¥ + + + = ¥ L 8. n odd : ( ) lim sgn n x a x b x c a fi¥ + + + = ¥ L 9. n odd : ( ) lim sgn n x a x c x d a fi-¥ + + + = - ¥ L Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Evaluation Techniques Continuous Functions If ( ) f x is continuous at a then ( ) ( ) lim x a f x f a fi = Continuous Functions and Composition ( ) f x is continuous at b and ( ) lim x a g x b fi = then ( ) ( ) ( ) ( ) ( ) lim lim x a x a f g x f g x f b fi fi = = Factor and Cancel ( )( ) ( ) 2 2 2 2 2 2 6 4 12 lim lim 2 2 6 8 lim 4 2 x x x x x x x x x x x x x fi fi fi - + + - = - - + = = = Rationalize Numerator/Denominator ( ) ( ) ( ) ( ) ( )( ) 2 2 9 9 2 9 9 3 3 3 lim lim 81 81 3 9 1 lim lim 81 3 9 3 1 1 18 6 108 x x x x x x x x x x x x x x x fi fi fi fi - - + = - - + - - = = - + + + - = = - Combine Rational Expressions ( ) ( ) ( ) ( ) 0 0 2 0 0 1 1 1 1 lim lim 1 1 1 lim lim h h h h x x h h x h x h x x h h h x x h x x h x fi fi fi fi ± ² - + ± ² - = ³ ´ ³ ´ ³ ´ + + L l L l ± ² - - = = = - ³ ´ ³ ´ + + L l L’Hospital’s Rule If ( ) ( ) 0 lim 0 x
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