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Quiz 7-3 to 7-5 Trig Integrals Trig Substitution (notecard) Partial Fractions True/False 1) A limit describes how a function moves as x moves toward a certain point. 2) A limit is a number or point past which a function cannot go. 3) A limit is a number that the y-values of a function can be made arbitrarily close to by restricting x-values. True/False 4) A limit is a number or point the functions gets close to but never reaches. 5) A limit is an approximation that can be made as accurate as you wish. 6) A limit is the value reached by plugging in numbers closer and closer to a given number. Indeterminate Forms and L’Hopital’s Rule Section 7.7 AP Calc Find the limit: 3x 12 lim x 2 x2 2 4x 7 lim 2 x 5 x 3 2 Thm 7.3 Extended Mean Value Theorem If f and g are differentiable on an open interval (a,b) and continuous on [a,b] such that g’(x)≠0 for any x in (a,b), then there exists a point c in (a,b) such that f ' (c) f (b) f (a) g ' (c) g (b) g (a) Thm 7.4 L’Hopital’s Rule Let f and g be functions that are differentiable on an open interval (a,b) containing c, except possibly at c itself. Assume g’(x)≠0 for all x in (a,b), except possibly at c itself. If the limit of f(x)/g(x) as x approaches c produces the indeterminate form 0/0, then f ( x) f ' ( x) lim lim x c g ( x ) x c g ' ( x ) f ( x) f ' ( x) lim lim x c g ( x ) x c g ' ( x ) provided the limit on the right exists (or is infinite). This result also applies if the limit of f(x)/g(x) as a approaches c produces any one of the indeterminate forms ∞/∞, (- ∞)/ ∞, ∞/(- ∞), or (- ∞)/(- ∞). Determine the limits: e (1 x) A) lim x 0 x x 2x 1 B) lim x 4 x 2 x ln x C) lim x x 2 4 x D) lim x e 2 x 3 Other indeterminate forms: 0 ,1 , ,0 , 0 0 Find the limit: 1 E) lim 1 x x x F) lim x 3 cot x x 0 x 8 G) lim 2 x2 x 4 x2 Use Graphing Calculator to check limits Notice: x x e e lim lim x 0 x x 0 1