790
Answer Key
2.13 . 1 3 2.14 . 1 4 2.15 . −1; 2.16 . 1 4 2.17 .
f ( x ) =−1
lim x →−1 −
2.18 . +∞ 2.19 . 0 2.20 . 0 2.21 . f is not continuous at 1 because f (1)=2≠3= lim x →1 f ( x ). 2.22 . f ( x ) is continuous at every real number. 2.23 . Discontinuous at 1; removable 2.24 . [−3, +∞) 2.25 . 0 2.26 . f (0) =1>0, f (1) =−2<0; f ( x ) is continuous over [0, 1]. It must have a zero on this interval. 2.27 . Let ε >0; choose δ = ε 3 ; assume 0< | x −2| < δ .
Thus,
| (3 x −2)−4 | = | 3 x −6 | = |3| · | x −2| <3· δ =3· ( ε /3) = ε . Therefore, lim x →2
3 x −2=4.
2.28 . Choose δ =min ⎧ ⎩
⎨ 9−(3− ε ) 2 , (3+ ε ) 2 −9 ⎫ ⎭ ⎬ .
2.29 . | x 2 −1 | = | x −1| · | x +1 | < ε /3·3= ε 2.30 . δ = ε 2 Section Exercises 1 . a. 2.2100000; b. 2.0201000; c. 2.0020010; d. 2.0002000; e. (1.1000000, 2.2100000); f. (1.0100000, 2.0201000); g. (1.0010000, 2.0020010); h. (1.0001000, 2.0002000); i. 2.1000000; j. 2.0100000; k. 2.0010000; l. 2.0001000 3 . y =2 x 5 . 3 7 . a. 2.0248457; b. 2.0024984; c. 2.0002500; d. 2.0000250; e. (4.1000000,2.0248457); f. (4.0100000,2.0024984); g. (4.0010000,2.0002500); h. (4.00010000,2.0000250); i. 0.24845673; j. 0.24984395; k. 0.24998438; l. 0.24999844 9 . y = x 4 +1 11 . π 13 . a. −0.95238095; b. −0.99009901; c. −0.99502488; d. −0.99900100; e. (−1;.0500000,−0;.95238095); f. (−1;.0100000,−0;.9909901); g. (−1;.0050000,−0;.99502488); h. (1.0010000,−0;.99900100); i. −0.95238095; j. −0.99009901; k. −0.99502488; l. −0.99900100
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online