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
x
a
f
x
L
fi
=
if
for every
0
e
>
there is a
0
d
>
such that
whenever
0
x
a
d
<
-
<
then
(
)
f
x
L
e
-
<
.
“Working” Definition :
We say
(
)
lim
x
a
f
x
L
fi
=
if we can make
(
)
f
x
as close to
L
as we want
by taking
x
sufficiently close to
a
(on either side
of
a
) without letting
x
a
=
.
Right hand limit :
(
)
lim
x
a
f
x
L
+
fi
=
. This has
the same definition as the limit except it
requires
x
a
>
.
Left hand limit :
(
)
lim
x
a
f
x
L
-
fi
=
. This has the
same definition as the limit except it requires
x
a
<
.
Limit at Infinity :
We say
(
)
lim
x
f
x
L
fi¥
=
if we
can make
(
)
f
x
as close to
L
as we want by
taking
x
large enough and positive.
There is a similar definition for
(
)
lim
x
f
x
L
fi-¥
=
except we require
x
large and negative.
Infinite Limit :
We say
(
)
lim
x
a
f
x
fi
= ¥
if we
can make
(
)
f
x
arbitrarily large (and positive)
by taking
x
sufficiently close to
a
(on either side
of
a
) without letting
x
a
=
.
There is a similar definition for
(
)
lim
x
a
f
x
fi
= -¥
except we make
(
)
f
x
arbitrarily large and
negative.
Relationship between the limit and one-sided limits
(
)
lim
x
a
f
x
L
fi
=
°
(
)
(
)
lim
lim
x
a
x
a
f
x
f
x
L
+
-
fi
fi
=
=
(
)
(
)
lim
lim
x
a
x
a
f
x
f
x
L
+
-
fi
fi
=
=
°
(
)
lim
x
a
f
x
L
fi
=
(
)
(
)
lim
lim
x
a
x
a
f
x
f
x
+
-
fi
fi
„
°
(
)
lim
x
a
f
x
fi
Does Not Exist
Properties
Assume
(
)
lim
x
a
f
x
fi
and
(
)
lim
x
a
g x
fi
both exist and
c
is any number then,
1.
(
)
(
)
lim
lim
x
a
x
a
cf
x
c
f
x
fi
fi
=
Ø
ø
º
û
2.
(
)
(
)
(
)
(
)
lim
lim
lim
x
a
x
a
x
a
f
x
g x
f
x
g x
fi
fi
fi
–
=
–
Ø
ø
º
û
3.
(
)
(
)
(
)
(
)
lim
lim
lim
x
a
x
a
x
a
f
x g x
f
x
g x
fi
fi
fi
=
Ø
ø
º
û
4.
(
)
(
)
(
)
(
)
lim
lim
lim
x
a
x
a
x
a
f
x
f
x
g x
g x
fi
fi
fi
Ø
ø
=
OE
oe
º
û
provided
(
)
lim
0
x
a
g x
fi
„
5.
(
)
(
)
lim
lim
n
n
x
a
x
a
f
x
f
x
fi
fi
Ø
ø
=
Ø
ø
º
û
º
û
6.
(
)
(
)
lim
lim
n
n
x
a
x
a
f
x
f
x
fi
fi
Ø
ø
=
º
û
Basic Limit Evaluations at
– ¥
Note :
(
)
sgn
1
a
=
if
0
a
>
and
(
)
sgn
1
a
= -
if
0
a
<
.
1.
lim
x
x
fi¥
= ¥
e
&
lim
0
x
x
fi- ¥
=
e
2.
(
)
lim ln
x
x
fi¥
= ¥
&
(
)
0
lim ln
x
x
-
fi
= -¥
3.
If
0
r
>
then
lim
0
r
x
b
x
fi¥
=
4.
If
0
r
>
and
r
x
is real for negative
x
then
lim
0
r
x
b
x
fi-¥
=
5.
n
even : lim
n
x
x
fi–¥
= ¥
6.
n
odd : lim
n
x
x
fi¥
= ¥
& lim
n
x
x
fi- ¥
= -¥
7.
n
even :
(
)
lim
sgn
n
x
a x
b x
c
a
fi–¥
+
+
+
=
¥
L
8.
n
odd :
(
)
lim
sgn
n
x
a x
b x
c
a
fi¥
+
+
+
=
¥
L
9.
n
odd :
(
)
lim
sgn
n
x
a x
c x
d
a
fi-¥
+
+
+
= -
¥
L
Calculus Cheat Sheet
Visit
http://tutorial.math.lamar.edu
for a complete set of Calculus notes.
©
2005 Paul Dawkins
Evaluation Techniques
Continuous Functions
If
(
)
f
x
is continuous at
a
then
(
)
(
)
lim
x
a
f
x
f
a
fi
=
Continuous Functions and Composition
(
)
f
x
is continuous at
b
and
(
)
lim
x
a
g x
b
fi
=
then
(
)
(
)
(
)
(
)
(
)
lim
lim
x
a
x
a
f
g x
f
g x
f
b
fi
fi
=
=
Factor and Cancel
(
)(
)
(
)
2
2
2
2
2
2
6
4
12
lim
lim
2
2
6
8
lim
4
2
x
x
x
x
x
x
x
x
x
x x
x
x
fi
fi
fi
-
+
+
-
=
-
-
+
=
=
=
Rationalize Numerator/Denominator
(
)
(
)
(
)
(
)
(
)(
)
2
2
9
9
2
9
9
3
3
3
lim
lim
81
81 3
9
1
lim
lim
81
3
9
3
1
1
18
6
108
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
fi
fi
fi
fi
-
-
+
=
-
-
+
-
-
=
=
-
+
+
+
-
=
= -
Combine Rational Expressions
(
)
(
)
(
)
(
)
0
0
2
0
0
1
1
1
1
lim
lim
1
1
1
lim
lim
h
h
h
h
x
x
h
h
x
h
x
h
x x
h
h
h
x x
h
x x
h
x
fi
fi
fi
fi
±
²
-
+
±
²
-
=
³
´
³
´
³
´
+
+
L
l
L
l
±
²
-
-
=
=
= -
³
´
³
´
+
+
L
l
L’Hospital’s Rule
If
(
)
(
)
0
lim
0
x
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- Spring '09
- Calculus, Derivative, Limits, lim, dx, Paul Dawkins
-
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