2.8. Existence of limits
In
general, no guarantee that a function f
(x)
actually has a limit as
.
If there is no limit, then we say that the limit
does not exist.
T
heorem:
A function f (x)
has a limit as x
approaches x0
if and only if
the right-hand and left-hand limit at
exist and are
equal. In symbols,
Remark:
Keep in mind that the symbols
and
are simply descriptions of limits that fail to exist. These symbols
do not represent real numbers and consequently they can not be
manipulated using rules of algebra. For example, it is not correct to
write
=0.
Example:
Figure 2.1 shows the
graph of function f whose domain is the closed interval
.
a) Does
exist?
b) Does
exist?
c) Does
exist?
Solution:
a) Inspection of the graph shows that
and
Although
the two one-sided limits exist, they are not equal. Thus,
does not exist. In short, “ f
does not have a limit as
”.
b) Inspection of the graph shows that
and
Thus, exists and is 3. The solid dot at (2,2) shows that
f
(2)=2. This
information, however, plays no role in our examination of the limit
of f (x)
as
.
c)
Inspection of the graph shows that
and
Thus,
exists and is 2. Incidentally, the fact that f
(3) is equal to 2
is irrelevant in determining
.
2.9. Continuity
Definition: A function f is said to be continuous at a point c if the following conditions are satisfied:
1. f (c) is defined
2.
exists
3.
If one or more of the conditions in this definition fails to hold, then f is called discontinuous at c and c is said to be a point of discontinuity of f . If f is continuous at all points of an open interval
(a , b), then f is said to be continuous on (a , b).
Example:
is discontinuous at 2, because
f (2) is undefined.
Example:
is
also discontinuous
at 2 because g (2) =3, and
,
so that
.
Example:
Show that
is a continuous function.
Solution: We can write f (x) as
is continuous if x>0
or x<0.
is identical to the polynomial and all polynomials are continuous
functions. Thus, x=0
is the only point that remains to be considered. At this point
,
so it remains to show that
Because the formula for f changes at 0, it will be helpful to consider the one-sided limits at 0 rather than the two- sided limit. We obtain:
and
Thus, (2) holds and is continuous at x=0.
Theorem: If functions f and g are continuous at c, then
a) f + g is continuous at c;
b) f - g is continuous at c;
c) f g is continuous at c;
d) f / g is continuous at c if g(c)0
and is discontinuous at c if g(c)=0.
Theorem: A rational function is continuous everywhere except at the points where the denominator is zero.
Example:
Where is
continuous?
Solution:
By theorem, the ratio is continuous everywhere except at the points
where the denominator is zero. Since solution of
are
x=2
and x=3,
h(x)
is continuous everywhere except at these two points.
Exercises
In exercises 1-13 find the limits.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
as
a)
and b)
13.
as
In
each of exercises 14 and 15 there is a graph of functions.
14. (See Fig.2.2). Decide which of given limits exist, and evaluate those which do.
; 
;
;
;
15. (See Fig.2.3)
;
;
;
16.
Graph
a)
Find
and
.
b)
Does
exist?
If so, what is that? If not, why not?
In exercises 17-22 find points of discontinuity, if any.
17.
18.
19.
20.
21.
22.
23. Find a value for the constant k, if possible, that will make the function continuous.
a)
; b)
24.
Let f
(x)
equal the least integer that is greater than or equal to x.
For instance, f
(3)
=3, f
(3.4)
=4, f
(3.9)
=4. This function is sometimes denoted
and called the ” ceiling of x”.
Graph the function and answer the questions.
a)
Does
exist? If so, what is it?
b)
Does
exist? If so, what is it?
c)
Does
exist? If so, what is it?
d) Is f continuous at 4?
e) Where is f continuous?
f) Where is f not continuous?
Answers
1. ; 2. ; 3. ; 4. ; 5. ; 6. ; 7. ; 8. ; 9. ; 10. ; 11. 0; 12. a) ; b) ; 13. a) ; b) ;
c) ; d) ; 14. a) 2; b) 1; c) 1; d) 3; 15. a) 2; b) 2; c) 1; d) 2;
16.
a) 1, 1; b) 1; 17.
none; 18.
none; 19.
;
20.
;
21. none; 22. none; 23. a) 5; b) 4/3; 24. a) yes; 4; b) yes; 5; c) no;
d) no; e) all nonintegers; f) all integers.