Calculus Limits - Cheat Sheet, Cheat Sheet of Calculus

Download Calculus Limits - Cheat Sheet and more Cheat Sheet Calculus in PDF only on Docsity! Calculus Cheat Sheet 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 a f x f x → →   =  Basic Limit Evaluations at ± ∞ Note : ( )sgn 1a = if 0a > and ( )sgn 1a = − if 0a < . 1. lim x x→∞ = ∞e & lim 0x x→− ∞ =e 2. ( )lim ln x x →∞ = ∞ & ( ) 0 lim ln x x +→ = −∞ 3. If 0r > then lim 0rx b x→∞ = 4. If 0r > and rx is real for negative x then lim 0rx b x→−∞ = 5. n even : lim n x x →±∞ = ∞ 6. n odd : lim n x x →∞ = ∞ & lim n x x →− ∞ = −∞ 7. n even : ( )lim sgnn x a x b x c a →±∞ + + + = ∞L 8. n odd : ( )lim sgnn x a x b x c a →∞ + + + = ∞L 9. n odd : ( )lim sgnn x a x c x d a →−∞ + + + = − ∞L