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AP Calculus Section 2.1 Limits (An Intuitive Approach) Homework: Page 110 #1 – 6 all and 7,9,11 Objective: SWBAT understand the basic concept of “limit.” The concept of a “limit” is the fundamental building block on which all calculus concepts are based. Students must understand this concept to fully appreciate calculus. Limits (An informal View) If the values of f ( x) can be made as close as we like to L by taking values of x sufficiently close to a (but not equal to a) then we write lim f ( x) L x a which is read “the limit of f ( x) as x approaches a is L” or “ f ( x) approaches L as x approaches a.” 1. The following table gives values for the function f ( x ) x (Radians) 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 .01 f ( x) sin x . x sin x x .84147 .87036 .89670 .92031 .94107 .95885 .97355 .98507 .99335 .99833 .99998 a. It appears that the function is approaching 1 when the values for x are approaching zero from the right. The notation for this would be lim x 0 sin x 1 x which is read “the limit of sin x as x approaches a from the right is 1” x b. It appears that the function is approaching 1 when the values for x are approaching zero from the left as well. The notation for this would be lim x 0 sin x 1 x which is read “the limit of sin x as x approaches a from the left is 1” x c. Since the limit from the right and the limit from the left are equal we can write sin x 1 x 0 x lim which is read “the limit of sin x as x approaches a is 1” x d. The notation in parts a and b above are called “ONE SIDED LIMITS” and the notation in part c is called a “TWO SIDED LIMIT”. A two-sided limit can only be used if the limit from both the left and the right are equal. The following notation summarizes the concept of a twosided limit. lim f ( x) L if and only if x a lim f ( x) lim f ( x) L x a x a The value of f at x a has no bearing on the limit as x approaches a. This is very important to remember. 2. Explain why lim x 0 x x does not exist. 3. Use the graph to find the limits. old book #2 a. c. e. lim f ( x) b. lim f ( x) d. f (2) lim f ( x) f. lim f ( x) x 2 x 2 x 4. . Use the graph to find the limits. lim f ( x) x 2 x Old book # 5 a. c. e. lim f ( x) b. lim f ( x) d. f (2) lim f ( x) f. lim f ( x) x 2 x 2 x 5. Limits at vertical asymptotes 6. Use the graph to find the limits. lim f ( x) x 2 x Old book #7 a. c. e. lim ( x) b. lim ( x) d. (2) lim ( x) f. lim ( x) x 2 x 2 x 7. The Greatest Integer Function lim ( x) x 2 x f ( x) x 12 10 8 6 4 2 -15 -10 -5 5 -2 -4 -6 -8 -10 -12 a. b. lim1 x x 2 lim x x 52 c. lim x d. lim x x2 x 0 10 15