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Cheat Sheet - Calculus Cheat Sheet Limits Definitions...

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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 xa f xL = if for every 0 e > there is a 0 d > such that whenever 0 <-< then ( ) f -< . “Working” Definition : We say ( ) lim f = if we can make ( ) fx as close to L as we want by taking x sufficiently close to a (on either side of a ) without letting = . Right hand limit : ( ) lim f + = . This has the same definition as the limit except it requires > . Left hand limit : ( ) lim f - = . This has the same definition as the limit except it requires < . Limit at Infinity : We say ( ) lim x f fi¥ = if we can make ( ) as close to L as we want by taking x large enough and positive. There is a similar definition for ( ) lim x f fi-¥ = except we require x large and negative. Infinite Limit : We say ( ) lim if we can make ( ) arbitrarily large (and positive) by taking x sufficiently close to a (on either side of a ) without letting = . There is a similar definition for ( ) lim = -¥ except we make ( ) arbitrarily large and negative. Relationship between the limit and one-sided limits ( ) lim f = & ( ) ( ) li m lim x a f x f +- fifi == ( ) ( ) li m lim x a f x f & ( ) lim f = ( ) ( ) li m lim x a f x & ( ) lim Does Not Exist Properties Assume ( ) lim and ( ) lim gx both exist and c is any number then, 1. ( ) ( ) li m lim x a c f xc = Øø ºû 2. ( ) ( ) ( ) ( ) li m li m lim x a x a f x g x f x =– 3. ( ) ( ) ( ) ( ) li m li m lim x a x a f xg x f x = 4. () ( ) lim lim lim g x = Œœ provided ( ) li m0 5. () () li m lim n n x a f x = 6. li m lim n n x a f x = Basic Limit Evaluations at –¥ Note : ( ) sg n1 a = if 0 a > and ( ) sg a =- if 0 a < . 1. lim x x fi¥ e & li x x = e 2. ( ) lim ln x x fi¥ & ( ) 0 lim ln x x - =-¥ 3. If 0 r > then li r x b x fi¥ = 4. If 0 r > and r x is real for negative x then li r x b x = 5. n even : lim n x x fi–¥ 6. n odd : lim n x x fi¥ & lim n x x 7. n even : ( ) li m sgn n x a x b x ca ++ + L 8. n odd : ( ) li m sgn n x a x b x fi¥ L 9. n odd : ( ) li m sgn n x a x c x da fi-¥ = L
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Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Evaluation Techniques Continuous Functions If ( ) fx is continuous at a then ( ) ( ) lim xa f x fa = Continuous Functions and Composition ( ) is continuous at b and ( ) lim g xb = then () ( ) () ( ) li m lim x a f g xf g x fb fifi == Factor and Cancel ( )( ) ( ) 2 2 22 2 26 4 12 li m lim 68 li m4 2 xx x x x x x -+ +- = -- + = Rationalize Numerator/Denominator ( ) ( ) ( ) ( ) ( ) 99 2 3 33 li m lim 8 1 81 3 91 li m lim 8 1 3 93 11 1 8 6 108 x xxx x x x - = + - + ++ - = =- Combine Rational Expressions ( ) ( ) ( ) ( ) 00 2 li m lim 1 li m lim hh x xh hxh x h xxh h h xx h hx -= ²³ Ll = = L’Hospital’s Rule If ( ) 0 lim 0 gx = or ( ) lim –¥ = then, ( ) ( ) li m lim x a f x g x ¢ = ¢ a is a number, ¥ or Polynomials at Infinity ( ) px and ( ) qx are polynomials. To compute ( ) lim x fi–¥ factor largest power of x in ( ) out of both ( ) and ( ) then compute limit. ( ) ( ) 2 2 2 2 2 2 4 4 5 5 3 3 3 43 li m li m lim 5 2 2 x x x x x x x x fi- ¥ ¥ fi-¥ - - - = = - Piecewise Function ( ) 2 lim x where 2 5 if 2 1 3 if 2 ´ + <- = µ - ‡- Compute two one sided limits, ( ) 2 li m li m 59 g - = += ( ) li m lim1 37 g - = One sided limits are different so ( ) 2 lim x doesn’t exist. If the two one sided limits had
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Cheat Sheet - Calculus Cheat Sheet Limits Definitions...

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