4.8
Limits at infinity
Definition 4.8.1. (a) Let f be defined on an interval (a, 1). A number L is the
limit of f as x approaches 1 if for every ✏ > 0 there exists a number M such
that
if x > M, then |f (x) L| < ✏.
In this case we write
lim f (x) = L,
x!1
and we say that f has a limit at 1 or the limit of f (x) exists as x approaches
1.
(b) Let f be defined on an interval ( 1, a). A number L is the limit of f as x
approaches 1 if for every ✏ > 0 there exists a number M such that
if x < M, then |f (x)
L| < ✏.
In this case we write
lim f (x) = L,
x! 1
and we say that f has a limit at
1.
1 or the limit of f (x) exists as x approaches
(c) If either limx!1 f (x) = L or limx! 1 f (x) = L, we call the horizontal line
y = L a horizontal asymptote of the graph of f .
Remark 4.8.1. If limx!1 f (x) and limx!1 g(x) both exist, then limx!1 (f (x) + g(x))
and limx!1 (f (x)g(x)) also exist and we have
lim (f (x) + g(x)) = lim f (x) + lim g(x),
x!1
x!1
x!1
and
lim (f (x)g(x)) = ( lim f (x))( lim g(x)).
x!1
Example 4.8.1.
(a)
x!1
limx!1 x1
= 0 and
limx! 1 x1
(b) For each positive integer n, limx!1
1
xn
x!1
= 0. Consequently,
= 0 and limx!
1
1 xn
= 0.
(c) If p(x) = an xn + an 1 xn 1 + an 2 xn 2 + . . . + a1 x + a0 and q(x) = bm xm +
bm 1 xm 1 + bm 2 xm 2 + . . . + b1 x + b0 with an 6= 0 and bm 6= 0, where m, n are
poistive integers.
p(x)
q(x)
p(x)
.
q(x)
If n = m, then limx!±1
to the graph of f (x) =
p(x)
q(x)
p(x)
.
q(x)
If n < m, then limx!±1
to the graph of f (x) =
=
an
bn
and the line y =
an
bn
is a horizontal asymptote
= 0 and the line y = 0 is a horizontal asymptote
If n > m the graph of f (x) =
p(x)
q(x)
has no horizontal asymptote.
21
Infinite limits at infinity
Definition 4.8.2. (a) Let f be defined on an interval (a, 1). If for any real number
N there is a real number M such that
if x > M, then f (x) > N,
we say that the limit of f (x) as x approaches 1 is 1 and we write
lim f (x) = 1.
x!1
(b) Let f be defined on an interval ( 1, a). If for any real number N there is a
real number M such that
if x < M, then f (x) < N,
we say that the limit of f (x) as x approaches
lim f (x) =
x! 1
(c) limx!1 f (x) =
1
1 and we write
1.
f (x) = 1 are defined in the same way.
(a) For any positive integer, limx!1 xn = 1.
⇢
1 if n is even
n
1x =
1 if n is odd
Example 4.8.2.
(b) limx!
1 and limx!
1 is
(c) If p(x) is a polynomial of degree at least 1, limx!1 p(x) = 1 or limx!1 p(x) =
1.
(d) If p(x) is a polynomial of degree at least 1, limx!
1.
(e) limx!1 ln x = 1, limx!1 ex = 1, limx!
4.9
1
1
p(x) = 1 or limx!
1
p(x) =
ex = 0.
Graphing
Definition 4.9.1. Let a function f be defined on a domain D. If for all x in D, we
have x in D and f ( x) = f (x), we say that the function f is even. In this case the
graph of f is symmetric with respect to the y axis.
If for all x in D, we have x in D and f ( x) = f (x), we say that the function
f is odd. In this case the graph of f is symmetric with respect to the the origin.
Remark 4.9.1. We can now combine all we have learned so far to draw the graph of
a function f . The following steps are helpful:
1. Find the domain of f .
22