Solução_Calculo_Stewart_6e

F.
TX.10
SECTION 2.2 THE LIMIT OF A FUNCTION ¤ 43
2.2 The Limit of a Function
1. As x approaches 2, f(x) approaches 5. [Or,thevaluesoff(x) can be made as close to 5 as we like by taking x sufficiently
close to 2 (but x 6= 2).] Yes, the graph could have a hole at (2, 5) and be definedsuchthatf(2) = 3.
3. (a) lim f(x) =∞ means that the values of f(x) canbemadearbitrarilylarge(aslargeasweplease)bytakingx
x→−3
(b)
sufficiently close to −3 (but not equal to −3).
lim f(x) =−∞ means that the values of f(x) can be made arbitrarily large negative by taking x sufficiently close to 4
x→4 +
through values larger than 4.
5. (a) f(x) approaches 2 as x approaches 1 from the left, so lim f(x) =2.
x→1− (b) f(x) approaches 3 as x approaches 1 from the right, so lim f(x) =3.
x→1 +
(c) lim
x→1
f(x) does not exist because the limits in part (a) and part (b) are not equal.
(d) f(x) approaches 4 as x approaches 5 from the left and from the right, so lim
x→5
f(x) =4.
(e) f(5) is not defined, so it doesn’t exist.
7. (a) lim g(t) =−1
t→0− (b) lim g(t) =−2
+
(c) lim
t→0
g(t) does not exist because the limits in part (a) and part (b) are not equal.
(d) lim g(t) =2
t→2− t→0
(e) lim g(t) =0
+
(f ) lim
t→2
g(t) does not exist because the limits in part (d) and part (e) are not equal.
(g) g(2) = 1
t→2
(h) lim
t→4
g(t) =3
9. (a) lim f(x) =−∞ (b) lim f(x) =∞
x→−7 x→−3
(d)
lim f(x) =−∞
x→6− (e) lim f(x) =∞
+
x→6
(f ) The equations of the vertical asymptotes are x = −7, x = −3, x =0,andx =6.
(c) lim f(x) =∞
x→0
11. (a) lim f(x) =1
x→0− (b)
lim f(x) =0
x→0 +
(c) lim
x→0
f(x) does not exist because the limits
in part (a) and part (b) are not equal.
13. lim f(x) =2, lim
x→1− x→1
f(x) =−2, f(1) = 2 15. lim f(x) =4, lim
+ +
x→3
f(3) = 3, f(−2) = 1
f(x) =2, lim f(x) =2,
x→3− x→−2