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

Section 2.2 The Limit of a Function 91
the four cases shown. In general, knowledge of vertical asymptotes is very useful in
sketching graphs.
y
5
Figure 15
2x
y=
x-3
0 x
x=3
ExamplE 9 Find lim
x l3 1
2x
and lim
x 2 3 x l3 2
2x
x 2 3 .
SOLUTION If x is close to 3 but larger than 3, then the denominator x 2 3 is a small
positive number and 2x is close to 6. So the quotient 2xysx 2 3d is a large positive
number. [For instance, if x − 3.01 then 2xysx 2 3d − 6.02y0.01 − 602.] Thus, intuitively,
we see that
lim
x l3 1
2x
x 2 3 − `
Likewise, if x is close to 3 but smaller than 3, then x 2 3 is a small negative number
but 2x is still a positive number (close to 6). So 2xysx 2 3d is a numerically large negative
number. Thus
lim
x l3 2
2x
x 2 3 − 2`
The graph of the curve y − 2xysx 2 3d is given in Figure 15. The line x − 3 is a vertical
asymptote.
■
y
ExamplE 10 Find the vertical asymptotes of f sxd − tan x.
SOLUTION Because
1
tan x − sin x
cos x
_
3π
2
_π
π
_
2
0
π
2
π
3π
2
x
there are potential vertical asymptotes where cos x − 0. In fact, since cos x l 0 1 as
x l sy2d 2 and cos x l 0 2 as x l sy2d 1 , whereas sin x is positive (near 1) when x
is near y2, we have
Figure 16
y − tan x
y
y=ln x
lim tan x − ` and lim tan x − 2`
x lsy2d2 x lsy2d1 This shows that the line x − y2 is a vertical asymptote. Similar reasoning shows
that the lines x − y2 1 n, where n is an integer, are all vertical asymptotes of
f sxd − tan x. The graph in Figure 16 confirms this.
■
0
1
x
Another example of a function whose graph has a vertical asymptote is the natural
logarithmic function y − ln x. From Figure 17 we see that
lim ln x − 2`
x l01 Figure 17
The y-axis is a vertical asymptote of
the natural logarithmic function.
and so the line x − 0 (the y-axis) is a vertical asymptote. In fact, the same is true for
y − log b x provided that b . 1. (See Figures 1.5.11 and 1.5.12.)
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