AB Calculus: Wks 1-3
LIMITS and CONTINUITY
Name ___________________
Limits
Informally, a limit is a y-value which a function approaches as x approaches some value.
lim f  x   L means as x approaches c, f  x  approaches the y-value of L.
x c
y  f  x
limits:
1. lim f  x  
function values:
8. f  4 
2. lim f  x  
9. f  1 
3. lim f  x  
10. f  2  
4. lim f  x  
11. f  3 

5. lim f  x  
12. f  5 

x  4
x 1
x 2
x 3
x 5
y


x






one-sided limits:
6. lim f  x  
x 5
7. lim f  x  
x 5
Continuity
Informally, a function is continuous where it can be drawn without lifting a pencil. Roughly,
continuous means “connected.”
Formally, a function is continuous where its limit and function value are the same.
In this course, we will work with three types of discontinuities: holes, vertical asymptotes, and
jumps (breaks).
All discontinuities can be classified as removable or nonremovable.
Removable discontinuities occur when the function has a limit (holes in the graph).
Nonremovable discontinuities occur when the limit of the function does not exist (jumps and
vertical asymptotes).
For piecewise functions, one-sided limit evaluation is often necessary.
 4  x, x  1
13. If f  x   
,
lim f  x  
2
x 1
4 x  x , x  1
3x  x3 , x  1
14. If g  x    2
,
lim g  x  
x 1
 2 x  1, x  1
15. For this same g function, lim g  x  
x 1
y
Another function requiring one-sided limit analysis is a step function
called the Greatest Integer Function. f ( x)  x the greatest
integer less than or equal to x. The graph is shown at the right.

x

16. lim1 x 
x
17.
lim x 


x 1
2
Intermediate Value Theorem
If f is continuous on  a, b and k is any y-value between
f  a  and f  b  , then there is at least one x-value c
between a and b such that f  c   k.
Examples:
18. Use the Intermediate Value Theorem to determine
if there would be a c-value on the given interval.
a. f  x   x 2  x,
b.
x2  4
g  x 
,
x2
f  c   12,  0,5
g  c   4,  0,3
f b
k
f a
a
c b
c. Find the value of c
 x3 , x  2
19. If the function f  x    2
is continuous, find the value of a.
ax , x  2
x  1
 x  1,

20. Find the values of a and b so that f  x   ax  b, 1  x  1 is continuous.
 2 x  1,
x 1

Use the Intermediate Value Theorem to determine if there would be a c-value on the
given interval. If there is a c find its value.
21. f  x   x2  x  1,
f  c   11,
0,5
x
,
x 1
23. f  x   x ,
f  c   1,
0, 2
f  c   3,
 4,1
22. f  x  
AP Calculus
Section ________
Name _______________________
Date ________________________
ASSIGNMENT 1-3
Use the graph of y  f  x  at the right to find these values.
1. lim f  x 
2.
4. lim f  x 
7. lim f  x 
x 5
x  3
x 0
lim f  x 
x 3
3. lim f  x 
5.
f  3
6. lim f  x 
8.
f  0
9. lim f  x 
x  3
x4
11. f  4 
12. f  2 
13. lim f  x 
14. f 1
15. lim f  x 
x2

x
x 4
10. lim f  x 
x 4
y







x 1
16. List the x-values of all removable discontinuities of f  x  .
17. List the x-values of all nonremovable discontinuities of f  x  .
Use the graph shown to find each value.
18. a. lim f  x 
x 0
c. f  0 
y

b. lim f  x 
f  x 
x2
x
d. removable discontinuities
2 x 2  4 x
2x

Use the graphs shown to find each value.
19. a. lim g  x 
x 0
b. lim g  x 
x  1
c. lim g  x 
x 1
f. g  0
d. lim g  x 
e. g  1
g. lim g  x 
h. removable discontinuities
x 1
x 2
y
i. nonremovable discontinuities

g  x 
1
x 1
x




20. a. h  1
b. h 1

c. lim h  x 
d. lim h  x 
e. lim h  x 
f. lim h  x 

x 1
x 1
x 1
y
x
x 1

g. removable discontinuities
h. nonremovable discontinuities
x  1
  x,

h  x    x  2, 1  x  1
 1,
x 1



21. a. lim f  x 
x3
c. lim f  x 
x  3
b. lim f  x 
f  x 
x 0
d. f  3
1
x 9
y
x
2





e. Use interval notation to show where f is continuous.
y
b. lim g  x 
22. a. g 1
c. lim g  x 
x 1
x 1
x

d. lim g  x 
x 1

g  x  x  2

e. removable discontinuities
f. nonremovable discontinuities

Find the indicated limits.

23. lim x 2  4
x 2
x
x  1 x  1
27. lim
2

24. lim  x 2  x 
x  3
28. lim
x  2
3
x2  4
25. lim x2  9
x0
29. lim
x 0
x
x 1
26. lim  2 x  5
x3
30. lim  3x  3
x 0
Use these piecewise functions to find the following.
31.
32.
 x 2  2, x  0
a. lim f  x  b. lim f  x  c. lim f  x 
f  x  
x 0
x 0
x 0
 x  2, x d.0 Give the intervals where f is continuous.
3x 2  x, x  2
g  x  
 2 x  6, x  2
a. lim g  x 
x 2
b. lim g  x 
x 0
c. List the interval(s) where g is continuous.
3