816
Answer Key
4.16 . f has a local minimum at −2 and a local maximum at 3. 4.17 . f has no local extrema because f ′ does not change sign at x =1. 4.18 . f is concave up over the interval ⎛ ⎝ −∞, 1 2 ⎞
⎠ and concave down over the interval ⎛ ⎝ 1
⎞ ⎠
2 , ∞
4.19 . f has a local maximum at −2 and a local minimum at 3. 4.20 . Both limits are 3. The line y =3 is a horizontal asymptote. 4.21 . Let ε >0. Let N = 1 ε . Therefore, for all x > N , we have | 3− 1 x 2
−3 | = 1 x 2
< 1
= ε Therefore,
N 2
lim x →∞ ⎠ =3. 4.22 . Let M >0. Let N = M ⎛ ⎝ 3−1/ x 2 ⎞
x > N , we have 3 x 2 >3 N 2 =3 ⎛
⎞ ⎠
2
2= 3 M
⎝ M 3
3 . Then, for all
M
3 =
4.23 . −∞ 4.24 . 3 5 4.25 . ± 3 4.26 . lim x →∞
f ( x ) =−2
lim x →−∞
f ( x ) = 3 5 ,
4.27 .
4.28 .
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