Calculus_Cheat_Sheet_All.pdfVIP

Calculus Cheat Sheet Visit for a complete set of Calculus notes. ? 2005 Paul Dawkins Limits Definitions Precise Definition : We say ( )lim x a f x L → = if for every 0ε there is a 0δ such that whenever 0 x a δ ? then ( )f x L ε? . “Working” Definition : We say ( )lim x a f x L → = 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 +→ = . This has the same definition as the limit except it requires x a . Left hand limit : ( )lim x a f x L ?→ = . This has the same definition as the limit except it requires x a . Limit at Infinity : We say ( )lim x f x L →∞ = 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 →?∞ = except we require x large and negative. Infinite Limit : We say ( )lim x a f x → = ∞ 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 → = ?∞ except we make ( )f x arbitrarily large and negative. Relationship between the limit and one-sided limits ( )lim x a f x L → = ? ( ) ( )lim lim x a x a f x f x L + ?→ → = = ( ) ( )lim lim x a x a f x f x L + ?→ → = = ? ( )lim x a f x L → = ( ) ( )lim lim x a x a f x f x + ?→ → ≠ ? ( )lim x a f x → Does Not Exist Properties Assume ( )lim x a f x → and ( )lim x a g x → both exist and c is any number then, 1. ( ) ( )lim lim x a x a cf x c f x → → =? ?? ? 2. ( ) ( ) ( ) ( )lim lim lim x a x a x a f x g x f x g x → → → ± = ±? ?? ? 3. ( ) ( ) ( ) ( )lim lim lim x a x a x a f x g x f x g x → → → =? ?? ? 4. ( ) ( ) ( ) ( ) lim lim lim x a x a x a f xf x g x g x → → → ? ? =? ? ? ? provided ( )lim 0 x a g x → ≠ 5. ( ) ( )lim lim nn x a x a f x f x → → ? ?=? ?? ? ? ? 6. ( ) ( )lim limn n x a x