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MATH 140
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Ewa Duma
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Lecture 10

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Mathematics & Statistics (Sci)

MATH 140

Ewa Duma

Summer

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2.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES
In Sections 2.2 and 2.4 we investigated inﬁnite limits and vertical asymptotes. There we
let x approach a number and the result was that the values of y became arbitrarily large
(positive or negative). In this section we let x become arbitrarily large (positive or nega-
x f▯x▯ tive) and see what happens to y. 132 |||| CHAPTER 2 LIMITS AND DERIVATIVES
0 ▯1 Let’s begin by investigating the behavior of the function f deﬁned by
2
▯1 0 f▯x▯ ▯40 - Lecture 10 Notes Again, the symbol ▯▯ does not represent a number, but the expression lim f▯x▯ ▯ L
▯2 0.600000 x ▯ 1 y=ƒ is often read as
▯3 0.800000
as x becomes large. The table at the left gives values of this function correct to six decimal “the limit of f▯x▯, as x approaches negative inﬁnity, is L”
▯4 0.882353 places, and the graph of f has been drawn by a computer in Figure 1.
▯5 0.923077 Limits at Inﬁnity and Horizontal Asymptotes:
▯10 0.980198 y y=L Deﬁnition 2 is illustrated in Figure 3. Notice that the graph approaches the line y ▯ L as
y=1 we look to the far left of each graph.
▯50 0.999200 0 x
▯100 0.999800
▯1000 0.999998 3
y DEFINITION The line y ▯ L is called a horizontal asymptote of the curve
0 1 ▯-1 xy ▯ f▯x▯ if either
y= ▯+1
FIGURE 1 lim f▯x▯ ▯ L
y=ƒ xl▯
y=L
As x grows larger and larger, the values of f(x) gets closer and closer to 1 but never to
to 1. This implies that we have a horizontal asymptote at 1 which means that the value willing
x sufﬁciently large. This situation is expressed symbolically by 0riting x For instance, the curve illustrated in Figure 1 has the line y ▯ 1 as a horizontal asymp-
get extremely close to 1 but never 1 itself. tote because
2
lim x ▯FIGUR▯ 1
Deﬁnition: Let f be a function dx l ▯x ▯ 1ome interval (a, inf). Then lim (x > inf) f(x) = L
Examples illustratixg_`im ƒ=L
which means that the values of f(x) can be made arbitrarily close to L by taking x
sufﬁciently large.
An example of a curve with two horizontal asymptotes is y ▯ tan x. (See Figure 4.)
y In fact,
▯
Deﬁnition: Let f be a function deﬁned on some interval (-inf, a). Then lim (x > -inf) f(x) =
L which means that the values of f(x) can be made arbitrarily close to L by taking x
sufﬁciently large negative. 0 4 lim tan x ▯ ▯
x xl▯▯
Deﬁnition: The line y = L is called a horizontal asymptote of the curve y = f(x) if either lim
▯
(x > inf) f(x) = L or lim (x > -inf) f(x) = L _2 so both of the lines y ▯ ▯ ▯2 and y ▯
the fact that the lines x ▯ ▯▯2 are vertical asymptotes of the graph of tan.)
Example: Find the inﬁnite limits, limitsFIGURE 4ity, and asymptotes for the function f whose
y=tan–!x EXAMPLE 1 Find the inﬁnite limits, limits at inﬁnity, and asymptotes for the function f
graph is shown:
whose graph is shown in Figure 5.
y SOLUTIONWe see that the values of f▯x▯ become large as x l ▯1 from both sides, so
2
Notice that f▯x▯ becomes large negative as x approaches 2 from the left, but large posi-
0 2 x tive as x approaches 2 from the right. So
lim f▯x▯ ▯ ▯▯
xl2▯
Thus both of the lines x ▯ ▯1 and x ▯ 2 are vertical asymptotes.
We can see that y = inf when x approachFIGURE 5d when x approaches -1 from both As x becomes large, it appears that f▯x▯ approaches 4. But as x decreases through
left and right side. Therefore the vertical asymptotes are when x = -1, 2.

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