1549901369-Elements_of_Real_Analysis__Denlinger_

222 Chapter 4 • Limits of Functions

Theorem 4.4.26 (Horizontal Asymptotes of Rational Functions) Con-
sider the rational function

R(x) = anxn + an-1Xn-l + · · · + a1x + ao.

bmxm + bm-1xm- l + · · · + bix + bo

(a) If n > m, then th e graph of R(x) has no horizontal asymptotes;

a

(b) If n = m, then the line y :== b n is a horizontal asymptote;

m

(c) If n < m, then the x-axis is a horizontal asymptote.

Proof. Exercise 19. •

~I ,

------------~----

'

y=l

------------/

x

f(x) = x + 2

x

/' y

Y! I

-----------r---y = 2 I

Figure 4.11

EXERCISE SET 4.4-B

1. Complete Definition 4.4.12.


2. Complete Definition 4.4.14.


3. Complete Definition 4.4.16

x = 2

--r------------I

'

: x

I I

I I 2x2

: f(x)= ---

' I x2 + x-6

I

4. Use Definitions 4.4.11- 4.4.16 to prove the following limit statements:

(a) lim 3x-1=~ (b) lim 3x2+2x-l=+oo

x-++oo 6x + 5 2 x-++oo X + 4

1-x^2 x - l

(c) lim --= +oo (d) lim --= 1

X->-00 X + 2 X->- 00 X + 1

(e) lim^1

1

• x

2

= -oo (f) lim x

2

+ x;^2 = -oo

x-++oo + X x-+-oo X +

5. Prove Theorem 4.4.18 (c).


6. Prove Theorem 4.4.21.