Calculus Cheat Sheet
Visit
http://tutorial.math.lamar.edu
for a complete set of Calculus notes.
© 2005 Paul Dawkins
Limits
Definitions
Precise Definition :
We say
( )
lim
xa
f
xL
ﬁ
=
if
for every
0
e
>
there is a
0
d
>
such that
whenever
0
<<
then
( )
f
<
.
“Working” Definition :
We say
( )
lim
f
ﬁ
=
if we can make
( )
fx
as close to
L
as we want
by taking
x
sufficiently close to
a
(on either side
of
a
) without letting
=
.
Right hand limit :
( )
lim
f
+
ﬁ
=
.
This has
the same definition as the limit except it
requires
>
.
Left hand limit :
( )
lim
f

ﬁ
=
.
This has the
same definition as the limit except it requires
<
.
Limit at Infinity :
We say
( )
lim
x
f
ﬁ¥
=
if we
can make
( )
as close to
L
as we want by
taking
x
large enough and positive.
There is a similar definition for
( )
lim
x
f
ﬁ¥
=
except we require
x
large and negative.
Infinite Limit :
We say
( )
lim
ﬁ
=¥
if we
can make
( )
arbitrarily large (and positive)
by taking
x
sufficiently close to
a
(on either side
of
a
) without letting
=
.
There is a similar definition for
( )
lim
ﬁ
= ¥
except we make
( )
arbitrarily large and
negative.
Relationship between the limit and onesided limits
( )
lim
f
ﬁ
=
&
( ) ( )
li
m
lim
x
a
f
x
f
+
ﬁﬁ
==
( ) ( )
li
m
lim
x
a
f
x
f
&
( )
lim
f
ﬁ
=
( ) ( )
li
m
lim
x
a
f
x
„
&
( )
lim
ﬁ
Does Not Exist
Properties
Assume
( )
lim
ﬁ
and
( )
lim
gx
ﬁ
both exist and
c
is any number then,
1.
( ) ( )
li
m
lim
x
a
c
f
xc
=
Øø
ºû
2.
( ) ( ) ( ) ( )
li
m
li
m
lim
x
a
x
a
f
x
g
x
f
x
ﬁ
–
=–
3.
( ) ( ) ( ) ( )
li
m
li
m
lim
x
a
x
a
f xg
x
f
x
ﬁ
=
4.
()
( )
lim
lim
lim
g
x
ﬁ
ﬁ
ﬁ
=
Œœ
provided
( )
li
m0
ﬁ
„
5.
() ()
li
m
lim
n
n
x
a
f
x
=
6.
li
m
lim
n
n
x
a
f
x
=
Basic Limit Evaluations at
–¥
Note :
( )
sg
n1
a
=
if
0
a
>
and
( )
sg
a
=
if
0
a
<
.
1.
lim
x
x
ﬁ¥
e
&
li
x
x
=
e
2.
( )
lim ln
x
x
ﬁ¥
&
( )
0
lim ln
x
x

ﬁ
=¥
3.
If
0
r
>
then
li
r
x
b
x
ﬁ¥
=
4.
If
0
r
>
and
r
x
is real for negative
x
then
li
r
x
b
x
=
5.
n
even : lim
n
x
x
ﬁ–¥
6.
n
odd : lim
n
x
x
ﬁ¥
& lim
n
x
x
7.
n
even :
( )
li
m
sgn
n
x
a
x
b
x
ca
++ +
L
8.
n
odd :
( )
li
m
sgn
n
x
a
x
b
x
ﬁ¥
L
9.
n
odd :
( )
li
m
sgn
n
x
a
x
c
x
da
ﬁ¥
=
¥
L
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View Full DocumentCalculus Cheat Sheet
Visit
http://tutorial.math.lamar.edu
for a complete set of Calculus notes.
© 2005 Paul Dawkins
Evaluation Techniques
Continuous Functions
If
( )
fx
is continuous at
a
then
( ) ( )
lim
xa
f
x
fa
ﬁ
=
Continuous Functions and Composition
( )
is continuous at
b
and
( )
lim
g
xb
ﬁ
=
then
()
( ) ()
( )
li
m
lim
x
a
f g
xf
g
x
fb
ﬁﬁ
==
Factor and Cancel
( )( )
( )
2
2
22
2
26
4
12
li
m
lim
68
li
m4
2
xx
x
x
x
x
x
ﬁ
+
+
=

+
=
Rationalize Numerator/Denominator
( )
( )
( )
( )
( )
99
2
3
33
li
m
lim
8
1
81 3
91
li
m
lim
8
1
3
93
11
1
8
6
108
x
xxx
x
x
x

=
+

+
++

=
=
Combine Rational Expressions
( )
( )
( )
( )
00
2
li
m
lim
1
li
m
lim
hh
x xh
hxh
x
h xxh
h
h xx
h
hx
&±
=
²³
Ll
=
=
L’Hospital’s Rule
If
( )
0
lim
0
gx
ﬁ
=
or
( )
lim
ﬁ
–¥
=
then,
( )
( )
li
m
lim
x
a
f
x
g
x
¢
=
¢
a
is a number,
¥
or
¥
Polynomials at Infinity
( )
px
and
( )
qx
are polynomials.
To compute
( )
lim
x
ﬁ–¥
factor largest power of
x
in
( )
out
of both
( )
and
( )
then compute limit.
( )
( )
2
2
2
2
2
2
4
4
5
5
3
3
3
43
li
m
li
m
lim
5
2
2
x
x
x
x
x
x
x
x
ﬁ
¥
¥
ﬁ¥



=
=

Piecewise Function
( )
2
lim
x
where
2
5
if
2
1
3
if
2
´
+
<
=
µ

‡
¶
Compute two one sided limits,
( )
2
li
m
li
m
59
g
ﬁ

=
+=
( )
li
m
lim1
37
g
ﬁ

=
One sided limits are different so
( )
2
lim
x
doesn’t exist. If the two one sided limits had
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 Spring '08
 Staff
 Calculus, Derivative, Limits, lim, dx, Paul Dawkins

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