Chapter Review 975 Limit properties (pp. 947–950) [ ] ( ) ( ) ( ) ( ) + = + → → → f x g x f x g x lim lim lim x c x c x c The limit of a sum equals the sum of the limits. [ ] ( ) ( ) ( ) ( ) − = − → → → f x g x f x g x lim lim lim x c x c x c The limit of a difference equals the difference of the limits. [ ] ( ) ( ) ( ) ( ) ⋅ = ⋅ → → → f x g x f x g x lim lim lim x c x c x c The limit of a product equals the product of the limits. ( ) ( ) ( ) ( ) ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = → → → f x g x f x g x lim lim lim x c x c x c ( ) ( ) ≠ → g x lim 0 x c The limit of a quotient equals the quotient of the limits, provided that the limit of the denominator is not zero. [ ] ( ) ( ) = ⎡ ⎣⎢ ⎤ ⎦⎥ → → f x f x lim lim x c n x c n Provided ( ) → f x lim x c exists, ≥ n 2 an integer. ( ) ( ) = → → f x f x lim lim x c n x c n Provided ( ) ( ) → f x f x and lim n x c n are both defined, ≥ n 2 an integer. Limit of a polynomial (p. 949) ( ) ( ) = → P x P c lim , x c where P is a polynomial function. One-sided Limits (p. 953–954) Left-hand limit: ( ) = → − f x L lim x c Right-hand limit: ( ) = → + f x R lim x c Continuous function (p. 956) A function f is continuous at c if ( ) ( ) = → f x f c lim . x c Derivative of a function (p. 962) ( ) ( ) ( ) ′ = − − → f c f x f c x c lim , x c provided the limit exists. Area under a graph (p. 972) If a function f is nonnegative and continuous on the interval [ ] a b , then the area under the graph of f from a to b is f x dx f u x lim , a b n i n i 1 ∫ ∑ ( ) ( ) = Δ →∞ = provided the limit exists. Section You should be able to . . . Example(s) Review Exercises 14.1 1 Investigate a limit using a table (p. 941) 1–4 1–11 2 Investigate a limit using a graph (p. 943) 5, 6 21–27 14.2 1 Find the limit of a sum, a difference, and a product (p. 947) 2–6 1 2 Find the limit of a polynomial (p. 949) 7 1, 2 3 Find the limit of a power or a root (p. 949) 8 2, 3, 5 4 Find the limit of a quotient (p. 950) 9–11 6–11 5 Find the limit of an average rate of change (p. 951) 12 30–32 14.3 1 Find the one-sided limits of a function (p. 953) 1 4, 21–24 2 Determine whether a function is continuous at a number (p. 955) 2, 3 12–15, 26–29 14.4 1 Find an equation of the tangent line to the graph of a function (p. 961) 1 30–32 2 Find the derivative of a function (p. 962) 2–4 33–37 3 Find instantaneous rates of change (p. 963) 5 39 4 Find the instantaneous velocity of an object (p. 964) 6 38 14.5 1 Approximate the area under the graph of a function (p. 969) 1, 2 40–42 2 Approximate integrals using a graphing utility (p. 972) 3 41(e), 42(e), 43(c), 44(c) Objectives Review Exercises In Problems 1–11, find the limit. 1. ( ) − + → x x lim 3 2 1 x 2 2 2. ( ) + →− x lim 1 x 2 2 2 3. + → x lim 7 x 3 2 4. − → − x lim 1 x 1 2 5. ( ) + → x lim 5 6 x 2 3 2 6. + + − →− x x x lim 2 9 x 1 2 2 7. − − → x x lim 1 1 x 1 3 8. − − − →− x x x lim 9 12 x 3 2 2 9. − − →− x x lim 1 1 x 1 2 3 10. − − + − → x x x x lim 8 2 4 8 x 2 3 3 2 11. − + − − + − → x x x x x x lim 3 3 3 2 6 x 3 4 3 3 2
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