Calculus_Cheat_Sheet_All

Calculus_Cheat_Sheet_All - Calculus Cheat Sheet Visit

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 f = if for every e > there is a d > such that whenever x a d <- < then ( ) f x L e- < . Working Definition : We say ( ) lim x a f x L f = 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 + f = . This has the same definition as the limit except it requires x a > . Left hand limit : ( ) lim x a f x L- f = . This has the same definition as the limit except it requires x a < . Limit at Infinity : We say ( ) lim x f x L f = 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 f- = except we require x large and negative. Infinite Limit : We say ( ) lim x a f x f = 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 f = - except we make ( ) f x arbitrarily large and negative. Relationship between the limit and one-sided limits ( ) lim x a f x L f = & ( ) ( ) li m lim x a x a f x f x L +- f f = = ( ) ( ) li m lim x a x a f x f x L +- f f = = & ( ) lim x a f x L f = ( ) ( ) li m lim x a x a f x f x +- f f & ( ) lim x a f x f Does Not Exist Properties Assume ( ) lim x a f x f and ( ) lim x a g x f both exist and c is any number then, 1. ( ) ( ) li m lim x a x a cf x c f x f f = 2. ( ) ( ) ( ) ( ) li m li m lim x a x a x a f x g x f x g x f f f = 3. ( ) ( ) ( ) ( ) li m li m lim x a x a x a f x g x f x g x f f f = 4. ( ) ( ) ( ) ( ) lim lim lim x a x a x a f x f x g x g x f f f = O o provided ( ) li m x a g x f 5. ( ) ( ) li m lim n n x a x a f x f x f f = 6. ( ) ( ) li m lim n n x a x a f x f x f f = Basic Limit Evaluations at Note : ( ) sg n 1 a = if a > and ( ) sg n 1 a = - if a < . 1. lim x x f = e & li m x x f- = e 2. ( ) lim ln x x f = & ( ) lim ln x x- f = - 3. If r > then li m r x b x f = 4. If r > and r x is real for negative x then li m r x b x f- = 5. n even : lim n x x f = 6. n odd : lim n x x f = & lim n x x f- = - 7. n even : ( ) li m sgn n x a x b x c a f + + + = L 8. n odd : ( ) li m sgn n x a x b x c a f + + + = L 9. n odd : ( ) li m sgn n x a x c x d a f- + + + =- 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 f = Continuous Functions and Composition ( ) f x is continuous at b and ( ) lim x a g x b f = then ( ) ( ) ( ) ( ) ( ) li m lim x a x a f g x f g x f b f f = = Factor and Cancel...
View Full Document

This note was uploaded on 03/13/2009 for the course MATH 2171 taught by Professor Deng during the Spring '08 term at UNC Charlotte.

Page1 / 11

Calculus_Cheat_Sheet_All - Calculus Cheat Sheet Visit

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online