Sec. 1.1 – 1.3
AP Calculus Practice Quiz – Limits
Name____________________
Date_____________________
Directions and Huge Hint: Don’t forget that you have four ways to examine every
mathematical problem. When symbols do not make sense, you should be graphing and
generating a number table as a matter of habit to try to figure out what is going on.
Talking it out with yourself also helps!
_______________________________________________________________________
1. Evaluate the limit numerically: lim
2ex  2
by creating a table that lists several
x
x 0
values of f(x) for values of x near 0.
Thus we conclude that lim
2ex  2
=
x
x 0
2. We can prove analytically that the limit in problem 1 exists, but not using the   
method. What other method of proof could we use? Hint: READ THE BOOK.
3. Another look at lim
x  0
sin x
: Notice from the number table in problem 1 that when x
x
values are “small”, sin x  x . This is a great example of
a) local linearity
d) a secant line
last update: 9-12-06 sw
b) instantaneous rate of change
e) none of these
c) average rate of change
1
Sec. 1.1 – 1.3
AP Calculus Practice Quiz – Limits
Name____________________
Date_____________________
4. Sketch the graph of the function and then use the graph to answer the questions in
parts a, b, and c. If the limit does not exist, write DNE and explain why the limit does not
exist.
f ( x) 
| x  1|
x 1
a) Find
lim f ( x)
x 1
b) Find lim f ( x)
x1
c) Find lim f ( x)
x2
5. Sketch the graph of the function and then use the graph to answer the questions in
parts a, b, and c. If the limit does not exist, write DNE and explain why the limit does not
exist.
f ( x) 
1
x
a) Find
lim f ( x)
x 0
b) (review): what are the asymptotes of f(x)?
last update: 9-12-06 sw
2
Sec. 1.1 – 1.3
AP Calculus Practice Quiz – Limits
Name____________________
Date_____________________
6. Sketch the graph of the function and then use the graph to answer the questions in
parts a, b, and c. If the limit does not exist, write DNE and explain why the limit does not
exist.
 x 2  2, x  1
f ( x)  
1, x  1
a) Find
lim f ( x)
x 0
b) lim f ( x)
x 1
7. Evaluate:
a) lim  =
x 0
b) lim 2 =
c) lim 0 =
x 0
x 2
Explain the meaning of each expression and your answer in a, b, and c using simple,
complete English sentences.
a)
b)
c)
last update: 9-12-06 sw
3
Sec. 1.1 – 1.3
AP Calculus Practice Quiz – Limits
Name____________________
Date_____________________
8. What were the four “break-through” problems of the 17th century that formed the basis
of contemporary calculus?
i)
ii)
iii)
iv)
9. Use the    definition of a limit to prove that lim (3x  5) exists.
x 2
10. Explain the basic underlying ideas or concepts of the    proof. A good
explanation might include both a graph as well as a verbal explanation.
11. Explain what the following means using an English sentence or sentences:
If lim f ( x)  L and if lim g ( x)  K , then lim[ f ( x) g ( x)]  LK .
x c
last update: 9-12-06 sw
x c
x c
4
Sec. 1.1 – 1.3
AP Calculus Practice Quiz – Limits
Name____________________
Date_____________________
x4 1
, if the limit exists. If it does not exist, writhe DNE and
x 1 x  1
explain why the limit does not exist.
12. Find the limit: lim
x 1 1
. Use an analytical (i.e., algebraic) method to alter the
x
rational function into an equivalent form.
13. Find the limit: lim
x 0
last update: 9-12-06 sw
5