216 Chapter 4 • Limits of Functions
- Prove Theorem 4.4.8 (d).
- Prove Theorem 4.4.10 (a).
- Prove Theorem 4.4.10 (b).
- Show by examples that the form ( +oo) + ( -oo) is indeterminate. That
is, in each of the following, find functions f and g satisfying the given
condition, and such that lim f(x) = +oo and lim g(x) = -oo:
X--+Xo X-+Xo
(a) lim [f(x) + g(x)] = 0 (b) lim [f(x) + g(x)] = +oo
x-+xo x-+xo
(c) lim [f(x) + g(x)J - oo (d) lim [f(x) + g(x)] = L =f. 0.
X-+Xo X-+Xo - Show by examples that the form ( +oo) · 0 is indeterminate. That is , in
each of the following, find functions f and g satisfying the given condition,
and such that lim f(x) = +oo and lim g(x) = 0:
x-+xo x-+xo
(a) lim f(x)g(x) = 0 (b) lim f(x)g(x) = +oo
X-+Xo X--+Xo
( c) lim f (x)g(x) = -oo (d) lim f(x)g(x) = L =f. 0.
X-+Xo X-+Xo - In each of the following, a function f and a number x 0 are given. Use the
approach of Example 4.4.4 to investigate lim f(x) and lim f(x), and
X->Xo X->xci
use the results to investigate lim f ( x).
x-+xo
x
(a) f(x) = (x _ l) 2 ; xo = 1
3
(b) f(x) = (x +
2
) 2 ; xo = -2
x
(c) f(x) = x _
1
; Xo = 1
x^2 -1
(e) f(x) = --; xo = 3
x-3x+3
(d) f(x) = x+
5
; xo = -5
x^2 + x - 2
(f) j(x) = ; xo = 2
x^2 - 3x + 2
3x - 9
(g) f(x) = ~
9; Xo = -3
x -(h) f ( x) = x
2
+
2
x +
1
; x o = 0
x^2 - 3x- Investigate each of the following (use the familiar algebraic properties of
cos x and sin x):
. 1
(a) hm cos -
X->0 X
(b) lim x cos .!.
x->O X
( c) lim .!. cos x
x->O- X
(d) lim .!. cos.!.
x->O+ X X
17. Vertical Asymptotes: The graph of a function f is said to have the
vertical line x = xo as a vertical asymptote if the domain of f contains
an interval of the form (xo - 8, x 0 ) or (x 0 , x 0 + 8) for some 8 > 0, and
f(x)---> +oo (or - oo) as x---> x 0 or x---> xt.