AP Calculus AB
Unit 1 Review - LIMITS
Name: ______________________________________
PART I - DO NOT USE A CALCULATOR ON ANY PROBLEM IN THIS SECTION. (Problems 1-29)
I. The graph of a function f is shown below.
6
f
4
2
-10
-5
5
10
-2
-4
Answer the following questions about function f.
1) f (5) 
2) f (2) 
3) f (4) 
4) lim f ( x) 
5) lim f ( x) 
6) lim f ( x) 
7) lim f ( x) 
8) lim f ( x) 
9) lim f ( x) 
10) lim f ( x) 
11) lim f ( x) 
12) lim f ( x) 
13) lim f ( x) 
14) lim f ( x) 
x 7
x4
x  0
x 
x 5
x 0
x  4
x2
x  0
x  4
x 
15) Use the definition of a continuous function at a number to answer the following.
a. f is not continuous at x = -7 because:_____________________________________________________
b. f is not continuous at x = 2 because:______________________________________________________
c. f is not continuous at x = 4 because:______________________________________________________
DO NOT USE A CALCULATOR
16) lim
-x2 + 4x
x→2
a) 0
17)
a)
lim
x9 
c) 4
d) -12
b) does not exist
c) 3
d)
e) None of these
x 3
x9
1
3
1
6
e) None of these
x
tan x
18) lim
x →0
19) lim
x →0
b) 12

1
1
x
b) - ∞ ∴does not exist
a) 1
c)
∞ ∴does not exist
d) 1
20) lim sinx
x →1
21)
True or False: If f is undefined at x = c, then the limit of f(x) as x approaches c does not exist.
e) None of these
22)
True of False: If the lim f(x)  L then f(c) = L.
23)
2

 x  3x  6 when x  2
lim f(x) when f(x) =  2
x →2

 x  3x  2 when x  2
xc
24)
Find a c such that f(x) is continuous on the entire real line.
x 2 when x  4
f(x) = 
 c when
x 4

x
25)
Find the x-values (if any) at which f is discontinuous. Label as removable or non-removable.
f(x) =
a)
b)
c)
d)
e)
2x  6
2x 2  18
x = 3 only….Non-removable
x = - 3 only…Non-removable
x = 3 and x = -3…Both non-removable
x = -3…Removable, x = 3…Non-removable
There are no discontinuities
26) Determine all of the vertical asymptotes of f(x):
x 2
f(x) = 2
x -4
a)
V.A at x = 2 only
27) If a  0, then lim
x  a
a)
1
6a2
x 2 - a2
x 4 - a4
b) V.A. at x = -2 only
c) V.A. at x = -2 and x = 2
d) No V.A’s
e) None of these
is:
b) 0
c)
1
a2
d)
1
2a2
e) Does not exist
28) If the function f is continuous on the closed interval [0, 2] and has values that are given in the table
below, then the equation f(x) = ½ must have at least TWO solutions in the interval [0, 2] if k = ? Hint! Draw a picture!
x
0
1
2
f(x)
1
k
2
a)
b)
c)
d)
e)
0
½
1
2
3
29) The graph of the function f is shown to the right.
Which of the following statements is false?
a)
b)
c)
d)
e)
x = a is in the domain of f.
lim f(x)
x a
is equal to
lim f(x) exists
x a
lim
x a
is not equal to
f is continuous at x = a
f(a)
lim f(x)
x a
AP Calculus AB
Unit 1 Review - LIMITS
Name: _________________________________
PART II - CALCULATORS MAY BE USED ON THE FIRST PART OF THIS SECTION. (Problems 1-6)
DO NOT USE A CALCULATOR FOR THE SECOND PART OF THIS SECTION (Problems 7-12)
1) Approximate the limit numerically by completing the table:
2
lim x = ___________
x 2
x
f(x)
2)
1.9
1.99
x -2
1.999
2
2.001
2.01
2.1
Find a function f(x) such that f(x) has a gap at x = 7 and a vertical asymptote at x = -4.
3) On the graph to the right, draw a function that has the following properties:
 A step (or jump) discontinuity at x = 5
 f(5) = 6.
4) Find the limit: lim
x0
(x  x)2  2(x  x)  1  (x 2  2x  1)
x
5) Create a function such that the lim does not exist because it is approaching +  from both the left and the right. Show
x 5
both the function and the graph.
NO CALCULATORS ON THIS SECTION. (Problems 7-6)
7) Find the limit:
lim
9) Find the limit:
lim
x
7  6 x5
x3
x
11) Find the limit: lim
x
8) Find the limit:
x2
2
x  2x 1
10) Find the limit:
5x2  2
x3
12) Find the limit:
lim
t
6  t3
7t 3  3
lim
r 
lim
y
5  2r 3
r2 1
2 y
7  6 y2