Calculus_Cheat_Sheet_All

Calculus_Cheat_Sheet_All - Calculus Cheat Sheet Limits...

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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 fx L = if for every 0 ε > there is a 0 δ > such that whenever 0 <−< then −< . “Working” Definition : We say ( ) lim = 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 x a = . Right hand limit : lim + = . This has the same definition as the limit except it requires x a > . Left hand limit : lim = . This has the same definition as the limit except it requires x a < . Limit at Infinity : We say lim x →∞ = 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 →−∞ = 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 x a = . There is a similar definition for ( ) lim = −∞ except we make ( ) arbitrarily large and negative. Relationship between the limit and one-sided limits lim = () ( ) lim lim +− →→ == ( ) ( ) lim lim = = ( ) lim = lim lim ( ) lim Does Not Exist Properties Assume lim and lim g x both exist and c is any number then, 1. () () lim lim cf x c f x = ⎡⎤ ⎣⎦ 2. () () () ( ) lim lim lim fx gx gx ±= ± 3. lim lim lim fxgx = 4. ( ) lim lim lim g xg x = ⎢⎥ provided ( ) lim 0 5. lim lim n n = 6. lim lim n n = Basic Limit Evaluations at ± ∞ Note : sgn 1 a = if 0 a > and sgn 1 a = − if 0 a < . 1. lim x x →∞ e & lim 0 x x →− ∞ = e 2. lim ln x x →∞ & 0 lim ln x x =−∞ 3. If 0 r > then lim 0 r x b x →∞ = 4. If 0 r > and r x is real for negative x then lim 0 r x b x = 5. n even : lim n x x →±∞ = ∞ 6. n odd : lim n x x →∞ = ∞ & lim n x x →− ∞ 7. n even : ( ) lim sgn n x ax bx c a + ++ = " 8. n odd : ( ) lim sgn n x a →∞ + = " 9. n odd : ( ) lim sgn n x cx d a →−∞ + = "
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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 fx fa = Continuous Functions and Composition is continuous at b and lim g xb = then lim lim fgx f gx fb →→ == Factor and Cancel 2 2 22 2 26 41 2 lim lim 68 lim 4 2 xx x x x x x −+ +− = −− + = Rationalize Numerator/Denominator ( ) 99 2 33 3 lim lim 81 81 3 91 lim lim 81 3 9 3 11 18 6 108 x x x x x + = + + + Combine Rational Expressions 00 2 1 1 lim lim 1 lim lim hh h hxh x h xxh h hxxh xxh x ⎛⎞ −= ⎜⎟ ++ ⎝⎠ = L’Hospital’s Rule If ( ) 0 lim 0 = or lim ±∞ = then, ( ) ( ) lim lim f x g xg x = a is a number, or −∞ Polynomials at Infinity ( ) px and ( ) qx are polynomials. To compute ( ) lim x →±∞ factor largest power of x out of both ( ) and ( ) and then compute limit.
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This note was uploaded on 03/29/2008 for the course MATH 251 taught by Professor Skrypka during the Spring '08 term at Texas A&M.

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Calculus_Cheat_Sheet_All - Calculus Cheat Sheet Limits...

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