BC 1
Limits 4
Name:
Limits and Continuity
The concept of limit allows us to replace our informal definition of continuity with one that is
mathematically precise.
Definition:
A function  is continuous at a point x = a in the domain of f if and only if
lim   x     a  .
xa
In the case f is continuous at every point in some set S, we say f is continuous on S.
If f is not continuous at x = a, we say the function is discontinuous at x = a.
(1)
(a)
Determine any point(s) of discontinuity (x-values) for each of the following functions.
x
(b)
  x 
 x  1 x  2
(c)
2
(2)
(3)
BC 1
1
g  x 
x
x 2  3x
3
 x2  9

, x3
Suppose   x    x  3
.
 4, x  3

(a)
Show that  is discontinuous at x = 3. Justify by checking each of the three
conditions in the definition for continuity listed above.
(b)
How should (3) be defined instead to make  continuous at that point?
(a)
Give an example of a function
with a removable discontinuity
(i.e., a hole) at x = 2.
(b)
Lim 4.1
Give an example of a function with a
jump discontinuity (i.e., a jump or a break
forming two separate sections) at x = 2.
(4)
(5)
(6)
(7)
BC 1
 3ax  7, x  1
Let h  x    2
. Find the value(s) of a so that h will be continuous at
 x  a, x  1
x  1 . (Use the definition and limits!)
Define the value of g ( 2) so that g will be continuous if g  x  
x 2  3x  10
x2  2 x
.
3x  2, x  2

x  2 , use the definition of continuity and limits to determine whether
If k  x    5,
 2
 x , x2
or not k is continuous at x = 2.
Find each limit. Approximate and guess if necessary.
ex
sin x
(a) lim
(b) lim
x  x
x
x
Lim 4.2
(c)
lim
x 
ln x
x
Intermediate Value Property: A function f satisfies the Intermediate Value Property (IVP) on the
interval [a,b] if and only if for each number k between f (a ) and f (b) , there exists a value
c  [ a, b] such that f (c )  k .
(8) Which of the following functions satisfy the IVP on the given interval?
a. The function below on [2,3] ?
2
1
b. The function f ( x ) 
c. Define f on (0,1) by
On [2, 0] ?
On [0, 3] ?
3
1
on [1,1] ?
x
On [2,10] ?
p
1
 if x is rational and x  in lowest terms
q
f ( x)   q
0 if x is irrational

.
1 3 
Does f satisfy the IVP on  ,
 .
2 2 
The Intermediate Value Theorem: If a function f is continuous on the closed interval [a, b] ,
then f satisfies the Intermediate Value Property (IVP) on the interval [a,b].
Alternately,
If a function f is continuous on the closed interval [a, b] , then for each number k between
f (a ) and f (b) , there exists a value c  [ a, b] such that f (c )  k .
BC 1
Lim 4.3