Calculus_Cheat_Sheet_All_Reduced

Calculus_Cheat_Sheet_All_Reduced - Calculus Cheat Sheet...

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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 gx both exist and c is any number then, 1. lim lim cf x c f x = ⎡⎤ ⎣⎦ 2. lim lim lim fx 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 →−∞ + ++= " Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Evaluation Techniques Continuous Functions If ( ) is continuous at a then ( ) ( ) lim fx fa = Continuous Functions and Composition ( ) is continuous at b and ( ) lim gx b = then lim lim fgx f 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 11 1 1 lim lim 1 lim lim hh xxh hxh x h xxh h hxxh x ⎛⎞ −= ⎜⎟ ⎝⎠ = L’Hospital’s Rule If ( ) 0 lim 0 = or ( ) lim = then, ( ) ( ) lim lim f x g 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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Calculus_Cheat_Sheet_All_Reduced - Calculus Cheat Sheet...

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