Intermediate Value TheoremSuppose that f x is continuous on and let M be any number between f a and f b. tan x and sec x provideddivision by zero.3 3 3. Rational function, except for x’s that give8. If the two one sided limits hadbeen equal then lim g x would have existedx 2and had the same value.Some Continuous FunctionsPartial list of continuous functions and the values of x for which they are continuous.1. x 2 3 423 423x 2 43xlim lim 2 5 lim 5 x x 5x 2 x2x x 2x x 2 x 2Piecewise Function 11 18 6 108Combine Rational Expressions1 11 1 x x h lim lim h 0 h x h x h 0 h x x h 1 h 11 lim lim h 0 x x h h 0 hx2 x x h p x factor largest power of x in q x outq x x 2 5 if x 2lim g x where g x x 2 1 3 x if x 2Compute two one sided limits,lim g x lim x 2 5 9x 2x 2x 2 x 2lim g x lim 1 3x 7One sided limits are different so lim g x x 2doesn’t exist. To computeFactor and Cancel x 2 x 6 x 2 4 x 12lim lim2x 2x 2x 2xx x 2 limx x 6 8 lim 4x 2x2Rationalize Numerator/Denominator3 x3 x 3 xlim 2 lim 2x 9 x 81x 9 x 813 x9 x 1 lim lim2x 9 x 81 3 x x 9 x 9 3 x of both p x and q x then compute limit. n odd : lim a x n c x d sgn a Visit for a complete set of Calculus notes.x 2005 Paul DawkinsĬalculus Cheat SheetEvaluation TechniquesL’Hospital’s RuleContinuous Functionsf x 0f x If f x is continuous at a then lim f x f a x aIf lim or lim then,x a g x x a g x 0 Continuous Functions and Compositionf x f x a is a number, or limlimf x is continuous at b and lim g x b thenx a g x x a g x x alim f g x f lim g x f b x ax aPolynomials at Infinityp x and q x are polynomials. If r 0 and x r is real for negative xbthen lim r 0x x3. lim ln x x lim e x 0x &lim ln x x 0 b 0xr4. lim n f x n lim f x x a x aBasic Limit Evaluations at Note : sgn a 1 if a 0 and sgn a 1 if a 0. lim x aprovided lim g x 0 x ax ag x g x limx ann5. lim f x g x lim f x lim g x x ax ax af x f x lim4. There is a similar definition for lim f x x aexcept we make f x arbitrarily large andsame definition as the limit except it requiresnegative.x a.Relationship between the limit and one-sided limitslim f x L lim f x lim f x Llim f x lim f x L lim f x Lx ax ax ax ax ax alim f x lim f x lim f x Does Not Existx a x ax aPropertiesAssume lim f x and lim g x both exist and c is any number then,x ax a1. This has thex ataking x large enough and positive.x x acan make f x arbitrarily large (and positive)by taking x sufficiently close to a (on either sideof a) without letting x a. This hasx athe same definition as the limit except itrequires x a. Infinite Limit : We say lim f x if wex aRight hand limit : lim f x L. can make f x as close to L as we want by“Working” Definition : We say lim f x LThere is a similar definition for lim f x Lif we can make f x as close to L as we wantexcept we require x large and taking x sufficiently close to a (on either sideof a) without letting x a. Calculus Cheat SheetLimitsDefinitionsLimit at Infinity : We say lim f x L if wePrecise Definition : We say lim f x L ifx ax for every 0 there is a 0 such thatwhenever 0 x a then f x L.
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