1549901369-Elements_of_Real_Analysis__Denlinger_

4.4 *Infinity in Limits 221

Theorem 4.4.24 (Limits of Polynomials at ±oo) Let p(x) = anxn +

an_ 1 xn-l + · · · + a 1 x + ao denote a polynomial. Then

1. Im p ( ) X = { +oo if an > O;

x-++oo -oo if an < 0.

If n IS. even , 1 Im. p ( ) x = { +oo if an > O;

X-+- 00 - CX) if an < 0.

If n IS. 0 dd ' 1 Im. p ( ) x = { -oo if an > O;

X-+-00 +oo if an < 0.

Proof. Exercise 18. •

y y

x x

n even nodd

f(x) = x4 -2x3 f(x) = i3 -3x + 3

Figure 4.9

RATIONAL FUNCTIONS AND HORIZONTAL ASYMPTOTES

Definition 4.4.25 The graph of a function f has a horizontal line y = m as a
horizontal asymptote if lim f(x) = m or lim f(x) = m.
x-++oo x-+- oo

y

y=m

------------------------------------------------------

x

Figure 4.10