## THE LIMIT

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## THE LIMIT

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- Pages:
- 4
- School:
- Whitman College
- Course:
- Math 125 - Calculus 1

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Calculus I Summary THE LIMIT 1 Definitions lim f x L lim f x L x a x a lim f x L x a 2 Algebraic Methods a Factor and Cancel b Multiply by Conjugate c Divide by xn Mainly for x d L Hospital s Rule For 0 0 or f 0 x f x lim 0 x a g x x a g x lim CANNOT be used to compute the derivative using limits i If form is 0 form f 1 g to get the 0 0 or form ii If form is f x g x rewrite so that f x g x eg x ln f x then take the limit of g x ln f x 3 Horizontal Vertical Asymptotes a x a is a vertical asymptote for f x if one of the following limits is infinite lim f x x a b y b is a horizontal asymptote for f x if one of the following is true lim f x b x 4 Heuristics that can be used a but is not necessarily 0 b If the denominator goes to zero but the numerator does not the limit is c If the denominator goes to and the numerator does not the overall limit goes to zero 5 Limit Laws Sect 2 3 6 Continuity a Definition lim f x f a x a Means 3 things 1 f a exists 2 The limit exists and 3 Items 1 and 2 are the same number b IVT If f is continuous on a b and w is a number between f a and f b there is at least one c in a b so that f c w Interpretations i If f is continuous the range of a closed interval is an interval ii If f is continuous and f x1 0 f x2 0 then there is a c between x1 and x2 where f c 0 f has at least one root in the interval between x1 and x2 c Continuous v Differentiable If f is differentiable at x a it is continuous at x a If f is continuous at x a we don t know if it is differentiable at x a That is All differentiable functions are continuous but not all continuous functions are differentiable 1 THE DERIVATIVE 1 Definition f 0 x lim h 0 f x h f x h 2 Rules for Differentiation See table 3 Higher derivatives a Differential Equations b Velocity and Acceleration c Inc Dec concave up down 4 Derivative of Inverses a Be able to derive the derivative formula for the inverse trig functions we needed implicit differentiation and triangles for this b If the point a b is on the

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