Page 48 - Calculus Workbook For Dummies

32 Part II: Limits and Continuity

^
^
For a function f xh and a real number a, lim f xh exists if and only if
x " a
^
1. lim f xh exists. In other words, there must be a limit from the left.
x " a -
^
2. lim f xh exists. There must be a limit from the right.
x " a +
^ h
^
3. lim f x = lim f xh The limit from the left must equal the limit from the right.
x " a - x " a +
(Note that this definition does not apply to limits as x approaches infinity or negative
infinity.)
^
And here’s the definition of continuity: A function f xh is continuous at a point x = a if
three conditions are satisfied:
^
1. f ah is defined.
^
2. lim f xh exists.
x " a
^ h
^
3. f a = lim f xh.
x " a
Using these definitions and Figure 3-1, answer problems 1 through 4.
y

15

10

Figure 3-1: 5
Graphus
interruptus:
A function 1 x
with many -5 -1 1 5 10 15
disconti-
nuities.