Page 55 - Schaum's Outline of Theory and Problems of Advanced Calculus

46 FUNCTIONS, LIMITS, AND CONTINUITY [CHAP. 3

2: lim ð f ðxÞ gðxÞÞ ¼ lim f ðxÞ lim gðxÞ¼ A B
x!x 0 x!x 0 x!x 0

3: lim ð f ðxÞgðxÞÞ ¼ lim f ðxÞ lim gðxÞ ¼ AB
x!x 0 x!x 0 x!x 0
A
lim f ðxÞ
x!x 0
4: lim if B 6¼ 0
f ðxÞ
B
¼
¼
lim gðxÞ
x!x 0 gðxÞ
x!x 0
Similar results hold for right- and left-hand limits.
INFINITY
It sometimes happens that as x ! x 0 , f ðxÞ increases or decreases without bound. In such case it is
customary to write lim f ðxÞ¼þ1 or lim f ðxÞ¼ 1, respectively. The symbols þ1 (also written
x!x 0 x!x 0
1) and 1 are read plus infinity (or infinity) and minus infinity, respectively, but it must be emphasized
that they are not numbers.
In precise language, we say that lim f ðxÞ¼1 if for each positive number M we can find a positive
x!x 0
number (depending on M in general) such that f ðxÞ > M whenever 0 < jx x 0 j < . Similarly, we say
that lim f ðxÞ¼ 1 if for each positive number M we can find a positive number such that
x!x 0
f ðxÞ < M whenever 0 < jx x 0 j < . Analogous remarks apply in case x ! x 0 þ or x ! x 0 .
Frequently we wish to examine the behavior of a function as x increases or decreases without bound.
In such cases it is customary to write x !þ1 (or 1)or x ! 1, respectively.
We say that lim f ðxÞ¼ l,or f ðxÞ! l as x !þ1,if for any positive number we can find a
x!þ1
positive number N (depending on in general) such that j f ðxÞ lj < whenever x > N. A similar
definition can be formulated for lim f ðxÞ.
x! 1

SPECIAL LIMITS

sin x 1 cos x
1. lim ¼ 1; lim ¼ 0
x!0 x x!0 x
x
1
2. lim 1 þ ¼ e, lim ð1 þ xÞ 1=x ¼ e
x
x!1 x!0þ
x
e 1 x 1
3. lim ¼ 1, lim ¼ 1
x!0 x x!1 ln x
CONTINUITY
Let f be defined for all values of x near x ¼ x 0 as well as at x ¼ x 0 (i.e., in a neighborhood of x 0 ).
The function f is called continuous at x ¼ x 0 if lim f ðxÞ¼ f ðx 0 Þ. Note that this implies three conditions
x!x 0
which must be met in order that f ðxÞ be continuous at x ¼ x 0 .
1. lim f ðxÞ¼ l must exist.
x!x 0
2. f ðx 0 Þ must exist, i.e., f ðxÞ is defined at x 0 .
3. l ¼ f ðx 0 Þ.
In summary, lim f ðxÞ is the value suggested for f at x ¼ x 0 by the behavior of f in arbitrarily small
x!x 0
neighborhoods of x 0 . Ifin fact this limit is the actual value, f ðx 0 Þ,of the function at x 0 , then f is
continuous there.
Equivalently, if f is continuous at x 0 ,wecan write this in the suggestive form lim f ðxÞ¼ f ð lim xÞ.
x!x 0 x!x 0