4.4 *Infinity in Limits 20913. Revise Theorem 4.2.20 to a correct theorem about limits from the left;
limits from the right.14. Revise Theorem 4.2.22 to a correct theorem about limits from the right.- Generalization of One-Sided Limits: Suppose f : A -r JR and g :
B -r JR, where A and B are disjoint and x 0 is a cluster point of both A
and B. Define h: AU B -r JR by h(x) = { f(x) ~f x EA}· Prove that
g(x) rfxEB
lim h(x) exists and equals Liff lim f(x) = lim g(x) = L.
X--+Xo X-+Xo X-+Xo4.4 *Infinity in Limits
This section is included here in the interest of completeness, to present a rigor-
ous justification of certain limit techniques involving "infinity." Most students
in this course have already gained a working knowledge of these concepts in
their elementary calculus courses. Thus, students may be encouraged to read
this section on their own if coverage is desired. The most significant items may
be the definitions and Theorems 4.4.3, 4.4.19, and 4.4.21.
INFINITY AS A LIMIT
Definition 4.4.1 Suppose f : D(f) -r JR, and Xo is a cluster point of D(f).
Then
(a) lim f(x) = +oo if
X--+Xo
'VM > 0, :JO> 0 3 'Vx E 'D(f), 0 < Ix - xol < O::::} f(x) > M.
(b) lim f(x) = -oo if
X-+Xo
'VM > 0, :Jo> 0 3 'Vx E 'D(f), 0 < Ix - xol < o::::} f(x) < -M.Note: If 'D(f) contains a deleted neighborhood of xo, then Definition 4.4.1
simplifies to:
lim f(x) = +oo if'VM > O,:Jo > 0 3 0 < lx-xol < o::::} f(x) > M;
X--+Xo
lim f(x) = -oo if'VM > O,:Jo > 0 3 0 < lx - xol < o::::} f(x) < -M.
X-+XQ
In words, lim f ( x) = +oo if for every M, f ( x) > M whenever x is sufficiently
X-+Xo
close to, but not equal to, xo. Similarly for lim f(x) = -oo.
X-+XQ
1
Example 4.4.2 Consider the limit statement lim (
2
x -+2 X - )^2 = +oo.
1
(a) Find o > 0 3 0 < Ix - 21 < o::::} (x _
2
) 2 > 1, 000.