Calculus Cheat Sheet Limits Definitions fx L fx fx lim x a lim x a e x a d 0e there is a L if 0d such that then Precise Definition We say for every whenever 0 Working Definition We say L f x as close to L as we want if we can make by taking x sufficiently close to a on either side of a without letting x Right hand limit the same definition as the limit except it requires x Left hand limit same definition as the limit except it requires x L This has the L This has lim x a lim x a a a a fx fx fx L lim x lim x L if we f x as close to L as we want by Limit at Infinity We say can make taking x large enough and positive fx There is a similar definition for except we require x large and negative Infinite Limit We say f x arbitrarily large and positive can make by taking x sufficiently close to a on either side of a without letting x There is a similar definition for except we make negative f x arbitrarily large and if we a lim x a lim x a f x f x Relationship between the limit and one sided limits L fxfx fxfx limlim limlim xax xax a f x fxf x Does Not Exist limlim xax lim x a cid 222 L a a cid 222 lim x a fx L Properties lim a x g x 4 both exist and c is any number then lim f x a x gxg x lim x a n fxf x fxf x limlim xax limlimn xax lim a x a a 6 5 n f x n provided lim a x g x 0 lim x a fx L cid 222 1 2 3 Assume and a f x lim x a cfxcf x limlim xax limlimlim xaxax limlimlim xaxax fxgxfxg x a fxgxfxg x a Basic Limit Evaluations at a if 1 a and x e 1 0 lim x Note 1 2 a if sgn lim x e x limln x 3 If r then lim 0 x limln x 0 0 sgn 0 x b r x x rx is real for negative x r and b 0 r x 0 then lim x 4 If x a 0 5 n even lim n x 6 n odd lim n x x 7 n even 8 n odd 9 n odd limsgn n x axbxc limsgnn x axcxd limsgn n x axbxc x L L L x lim n Visit http tutorial math lamar edu for a complete set of Calculus notes a a a 2005 Paul Dawkins Calculus Cheat Sheet Evaluation Techniques lim x a fxf a f x is continuous at a then Continuous Functions If Continuous Functions and Composition f x is continuous at b and fgxfgxf b limlim xax a Factor and Cancel gx lim x a b then x x 412 2 limlim xxx x 2 2 x 2 x x 2 lim x 2 x x 2 6 x 2 8 2 6 4 Rationalize Numerator Denominator f x g x then a is a number or 0 0 lim x a L Hospital s Rule or If f x lim g x a x x fxf limlim gxg x xax a Polynomials at Infinity p x and p x lim q x x p x and of both x 34 2 limlimlim x 522 xx 2 x 2 x q x are polynomials To compute factor largest power of x in q x out 2 q x then compute limit x x 4 2 x 3 5 x 4 2 x 2 3 5 x 2 3 33 limlim x 2 x 9 x x 2 x xx 3 8181 x 9 9 8139 3 1 9 limlim xxx 2 x 1 186108 3 x x x 9 1 0 Combine Rational Expressions xx h 1 111 cid 230 cid 246 limlim cid 231 cid 247 hxhxhxx h h cid 230 h 11 limlim cid 231 cid 231 hxxhxxh h 1 cid 230 cid 231 cid 231 cid 246 cid 247 cid 247 x h h 0 0 0 cid 246 cid 247 cid 247 2 2 2 x x 2 g x gx gx where Piecewise Function g x x 2 x 2 5if cid 236 lim cid 237 x 13if x 2 cid 238 Compute two one sided limits limlim5 9 x limlim13 x One sided limits are different so doesn t exist If the two one sided limits had g x been equal then would have existed and had the same value x x lim x 2 lim x 2 g x 7 x 2 2 2 Some Continuous Functions Partial list of continuous functions and the values of x for which they are continuous 1 Polynomials for all x sin x for all x 2 Rational function except for x s that give sec x provided 7 8 division by zero n x n odd for all x n x n even for all xe for all x ln x for x 0 x 0 cos x and tan x and 3 ppp p L 222 2 cot x and x 2 0 2 ppp p L x L csc x provided L 3 9 3 4 5 6 f x is continuous on a b and let M be any number between f a and f b Intermediate Value Theorem Suppose that Then there exists a number c such that ac b and fc M Visit http tutorial math lamar edu for a complete set of Calculus notes 2005 Paul Dawkins Calculus Cheat Sheet Derivatives Definition and Notation fxhf x f x lim h 0 h yf x all of the following are equivalent If notations for derivative evaluated at x fayDf a a x a dfdy dxdx xax a Interpretation of the Derivative f a change of 2 is the instantaneous rate of f x at x a 3 If f x is the position of an object at time x then the object at x f a a is the velocity of then the derivative is defined to be yf x If If equivalent notations for the derivative fxyfxDf x then all of the following are yf x d dfdy dxdxdx If yf x 1 is the slope of the tangent then yf x mf a line to equation of the tangent line at x given by yfafax a at x a and the a is f x and If x 1 cfcf fgfxg x 2 fgfgf g 3 ffgf g cid 246 cid 247 g 4 cid 230 cid 231 g 2 Product Rule Quotient Rule Basic Properties and Formulas g x are differentiable functions the derivative exists c and n are any real numbers 5 6 7 n n d c 0 dx d xn x dx d fgxfgxg x dx This is the Chain Rule 1 Power Rule Common Derivatives csccsccot xx x x cotcsc 1 sin x …