James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 2.6 Limits at Infinity; Horizontal Asymptotes 135
Figure 15 shows that if a smaller value of « is chosen, then a larger value of N may
be required.
L
y=L+∑
y=L-∑
y=ƒ
FIGURE 15
lim f sxd − L
x l `
0 x
N
Similarly, a precise version of Definition 2 is given by Definition 8, which is illustrated
in Figure 16.
8 Definition Let f be a function defined on some interval s2`, ad. Then
lim f sxd − L
x l 2`
means that for every « . 0 there is a corresponding number N such that
if x , N then | f sxd 2 L | , « x
y
y=ƒ
y=L+∑
L
y=L-∑
FIGURE 16
lim f sxd − L
x l2`
N
0
In Example 3 we calculated that
lim
x l `
3x 2 2 x 2 2
5x 2 1 4x 1 1 − 3 5
In the next example we use a graphing device to relate this statement to Definition 7 with
L − 3 5 − 0.6 and « − 0.1.
TEC In Module 2.4y2.6 you can
explore the precise definition of a limit
both graphically and numerically.
ExamplE 13 Use a graph to find a number N such that
if x . N then Z
3x 2 2 x 2 2
5x 2 1 4x 1 1 2 0.6 Z , 0.1
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