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

Section 2.6 Limits at Infinity; Horizontal Asymptotes 137
y
M
y=M
Finally we note that an infinite limit at infinity can be defined as follows. The geometric
illustration is given in Figure 19.
FIGURE 19
lim
x l ` f sxd − `
0 N x
9 Definition of an Infinite Limit at Infinity Let f be a function defined on
some interval sa, `d. Then
lim f sxd − `
x l `
means that for every positive number M there is a corresponding positive number
N such that
if x . N then f sxd . M
Similar definitions apply when the symbol ` is replaced by 2`. (See Exercise 80.)
1. Explain in your own words the meaning of each of the
following.
(a) lim
x l `
f sxd − 5
(b) lim
x l 2` f sxd − 3
1 x
1
2. (a) Can the graph of y − f sxd intersect a vertical asymptote?
Can it intersect a horizontal asymptote? Illustrate by
(e) lim x l2 1 (f) The equations of the asymptotes
sketching graphs.
(b) How many horizontal asymptotes can the graph of
y − f sxd have? Sketch graphs to illustrate the possibilities.
3. For the function f whose graph is given, state the following.
(a) lim f sxd
(b) lim 5–10 Sketch the graph of an example of a function f that satisfies
all of the given conditions.
x l` x l2`
(c) lim f sxd
(d) lim f sxd
x l1 x l3
5. lim f sxd − 2`,
(e) The equations of the asymptotes
f sxd − 5, lim f sxd − 25
x l 0 x l2` x l`
6. lim f sxd − `,
y
x l 2 lim f sxd − `,
x l221 lim f sxd − 2`,
x l222 lim f sxd − 0, lim f sxd − 0, f s0d − 0
x l2` x l`
1
7. lim f sxd − 2`,
x l2 lim f sxd − `,
x l` lim f sxd − 0,
x l2`
1 x
lim f sxd − `,
x l01 lim f sxd − 2`
x l02 8. lim f sxd − 3, lim f sxd − `, lim f sxd − 2`, f is odd
x l ` x l22 x l21 9. f s0d − 3, lim f sxd − 4,
x l02 lim f sxd − 2,
x l01 4. For the function t whose graph is given, state the following.
lim f sxd − 2`,
x l2` lim f sxd − 2`,
x l 42 lim f sxd − `,
x l 41 (a) lim tsxd
x l`
(b) lim x l2`
lim x l`
(c) lim tsxd
x l 0
(d) lim x l 2 2 10. lim f sxd − 2`, lim f sxd − 2,
x l 3 x l`
f s0d − 0, f is even
y
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