By Maxine Lifshitz
Offers an entire assessment of themes incorporated within the collage Board AP Calculus AB and Calculus BC direction descriptions and may aid scholars advance the abilities and methods had to prevail on either the Calculus AB and Calculus BC assessments.
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Additional info for Amsco's AP Calculus AB BC: Preparing for the Advanced Placement Exams
X ϩ 5 10 EXAMPLE 6 Evaluate lim x→a "x Ϫ "a xϪa . 0 Solution Substitute x ϭ a into the expression. The result is , an inde0 terminate form. Rationalize the numerator by multiplying the numerator and the denominator by the conjugate "x ϩ "a. a "x ϩ "a "x Ϫ "a xϪa 1 b xϪa ϭ ϭ "x ϩ "a x Ϫ a( "x ϩ "a) "x ϩ "a We then have lim x→a 44 Chapter 3 • Limits and Continuity 1 ϭ 1 . "x ϩ "a 2 "a EXAMPLE 7 Evaluate lim x→3 xϪ3 . 1Ϫ1 x 3 0 Solution Substituting x ϭ 3 into the expression gives the result , an 0 indeterminate form.
Find f(2) for f(x) ϭ b (A) (B) (C) (D) (E) x2 ϩ 4, x 6 0 . 3 Ϫ x, x Ն 0 1 3 4 8 does not exist 23. Which of the following graphs shows a function that is continuous for all real numbers? (A) y x y (B) Ϫ 4x ϩ 4 ϭ 4x2 Ϫ 1 (A) 1 4 (B) 1 (C) 4 (D) 8 (E) does not exist 20. Find the equation of the vertical asymptote x of f(x) ϭ . 4x ϩ 8 (A) y ϭ 1 4 (B) y ϭ Ϫ2 (C) x ϭ Ϫ2 (D) y ϭ 2 (E) x ϭ 2 56 Chapter 3 • Limits and Continuity x (C) y x (D) 3. On the graph of f(x), sketch the graph of fϪ1(x) and find f( fϪ1(Ϫ1)).
The equation of the horizontal asymptote is y ϭ b. Another way of expressing the fact that f(x) approaches the value b as x S q is to use the limit symbol: lim f(x) ϭ b xSq 39 The limit symbol can also be used to describe the end behavior of a function. 1, the end behavior of polynomial functions was discussed. In general, if a function f(x) is unbounded, this means that f(x) approaches q or Ϫq as x increases (as x S +q ) or as x decreases (as x S Ϫq ). If f(x) approaches q as x S q , the limit symbol expresses this as follows: lim f(x) ϭ q .