4.4 *Infinity in Limits 217. p(x).
(a) Prove that if R(x) = q(x), where p(x) and q(x) are polynomials,
then the graph of R(x) has a vertical asymptote x = x 0 at any point
xo where q(xo) = 0 but p(xo) -/= 0.
(b) Find all vertical asymptotes of the graph for each function given in
Exercise 15.
LIMITS AT INFINITYWe now consider limits as x-+ +oo or -oo. Six definitions are needed. We
supply three of them and leave the other three as exercises.
Definition 4.4.11 lim f(x) = L ¢? D(f) is unbounded above, and
x-++oo
Ve> 0, :JN> 0 3 't/x E D(f), x > N::::} lf(x) -LI < c.
Definition 4.4.12 lim f(x) = +oo ¢? ... (Exercise 1.)
x-++oo
Definition 4.4.13 lim f(x) = -oo ¢? D(f) is unbounded above, and
x-++oo
't/M > 0, :JN> 0 3 't/x E D(f), x > N::::} f(x) < -M.
Definition 4.4.14 lim f(x) = L ¢? ... (Exercise 2.)
x-+- oo
Definition 4.4.15 lim f(x) = +oo ¢? D(f) is unbounded below, and
X-+-00
't/M > 0, :JN> 0 3 't/x E D(f), x < -N::::} f(x) > M.
Definition 4.4.16 lim f(x) = -oo ¢? ... (Exercise 3.)
x-+-oo
Example 4.4.17 Prove that lim (5 - 4x) = -oo.
x-++oo
Solution: Let M > 0. Choose N = M:
5. Then
M+5
x>N::::}x>-
4- ::::} 4x > M +5
::::} 4x - 5 > M
::::} 5-4x < -M.
Therefore, by Definition 4.4.13, lim (5 - 4x) = -oo. 0
x-++oo