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
fxL
®
= if
for every 0
e
>
there is a 0
d
>
such that
whenever 0 xa
d
<-< then
(
)
fxL
e
-<.
“Working” Definition : We say
(
)
lim
xa
fxL
®
=
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
xa
=
.
Right hand limit :
(
)
lim
xa
fxL
+
®
=. This has
the same definition as the limit except it
requires
xa
>
.
Left hand limit :
(
)
lim
xa
fxL
-
®
=. This has the
same definition as the limit except it requires
xa
<
.
Limit at Infinity : We say
(
)
lim
x
fxL
®¥
= if we
can make
(
)
fx as close to L as we want by
taking x large enough and positive.
There is a similar definition for
(
)
lim
x
fxL
®-¥
=
except we require x large and negative.
Infinite Limit : We say
(
)
lim
xa
fx
®
=¥ if we
can make
(
)
fx arbitrarily large (and positive)
by taking x sufficiently close to a (on either side
of a) without letting
xa
=
.
There is a similar definition for
(
)
lim
xa
fx
®
=-¥
except we make
(
)
fx arbitrarily large and
negative.
Relationship between the limit and one-sided limits
(
)
lim
xa
fxL
®
=
Þ
(
)
(
)
limlim
xaxa
fxfxL
+-
®®
==
(
)
(
)
limlim
xaxa
fxfxL
+-
®®
==
Þ
(
)
lim
xa
fxL
®
=
(
)
(
)
limlim
xaxa
fxfx
+-
®®
¹
Þ
(
)
lim
xa
fx
®
Does Not Exist
Properties
Assume
(
)
lim
xa
fx
®
and
(
)
lim
xa
gx
®
both exist and c is any number then,
1.
(
)
(
)
limlim
xaxa
cfxcfx
®®
=éù
ëû
2.
(
)
(
)
(
)
(
)
limlimlim
xaxaxa
fxgxfxgx
®®®
±=±éù
ëû
3.
(
)
(
)
(
)
(
)
limlimlim
xaxaxa
fxgxfxgx
®®®
=éù
ëû
4.
()
()
(
)
()
lim
lim
lim
xa
xa
xa
fx
fx
gxgx
®
®
®
éù
=
êú
ëû
provided
(
)
lim0
xa
gx
®
¹
5.
() ()
limlim
n
n
xaxa
fxfx
®®
éù
=éù
ëû
ëû
6.
() ()
limlim
n
n
xaxa
fxfx
®®
éù
=
ëû
Basic Limit Evaluations at
±¥
Note :
(
)
sgn1a= if 0a
>
and
(
)
sgn1a=- if 0a
<
.
1. lim
x
x®¥
=¥e & lim0
x
x®-¥
=e
2.
(
)
limln
x
x
®¥
=¥ &
(
)
0
limln
x
x
+
®
=-¥
3. If 0r
>
then lim0
r
x
b
x
®¥
=
4. If 0r
>
and
r
x is real for negative x
then lim0
r
x
b
x
®-¥
=
5. n even : lim
n
x
x
®±¥
=¥
6. n odd : lim
n
x
x
®¥
=¥ & lim
n
x
x
®-¥
=-¥
7. n even :
(
)
limsgn
n
x
axbxca
®±¥
+++=¥
L
8. n odd :
(
)
limsgn
n
x
axbxca
®¥
+++=¥
L
9. n odd :
(
)
limsgn
n
x
axcxda
®-¥
+++=-¥
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
(
)
fxis continuous at a then
(
)
(
)
lim
xa
fxfa
®
=
Continuous Functions and Composition
(
)
fx is continuous at b and
(
)
lim
xa
gxb
®
= then
()
( )
()
(
)
()
limlim
xaxa
fgxfgxfb
®®
==
Factor and Cancel
(
)
(
)
( )
2
2
22
2
26
412
limlim
22
68
lim4
2
xx
x
xx
xx
xxxx
x
x
®®
®
-+
+-
=
--
+
===
Rationalize Numerator/Denominator
( )
( )
( )
( )
( )()
22
99
2
99
333
limlim
8181
3
91
limlim
81393
11
186108
xx
xx
xxx
xx
x
x
xxxx
®®
®®
--+
=
--
+
--
==
-+++
-
==-
Combine Rational Expressions
(
)
( )
( ) ( )
00
2
00
1111
limlim
111
limlim
hh
hh
xxh
hxhxhxxh
h
hxxhxxhx
®®
®®
æö
-+
æö
-=
ç÷
ç÷
ç÷
++
èø
èø
æö
--
===-
ç÷
ç÷
++
èø
L’Hospital’s Rule
If
(
)
()
0
lim
0
xa
fx
gx
®
= or
(
)
()
lim
xa
fx
gx
®
±¥
=
±¥
then,
(
)
()
(
)
()
limlim
xaxa
fxfx
gxgx
®®
¢
=
¢
a is a number,
¥
or
-¥
Polynomials at Infinity
(
)
px and
(
)
qx are polynomials. To compute
(
)
()
lim
x
px
qx
®±¥
factor largest power of x in
(
)
qxout
of both
(
)
px
and
(
)
qx then compute limit.
(
)
( )
2
2
2
2
2
2
4
4
5
5
3
3
343
limlimlim
5222
2
xxx
x
x
x
x
x
x
xx
x
®-¥®-¥®-¥
-
-
-
===-
--
-
Piecewise Function
(
)
2
lim
x
gx
®-
where
()
2
5if 2
13if 2
xx
gx
xx
ì
+<-
=
í
-³-
î
Compute two one sided limits,
(
)
2
22
limlim59
xx
gxx
--
®-®-
=+=
(
)
22
limlim137
xx
gxx
++
®-®-
=-=
One sided limits are different so
(
)
2
lim
x
gx
®-
doesn’t exist. If the two one sided limits had
been equal then
(
)
2
lim
x
gx
®-
would have existed
and had the same value.
Some Continuous Functions
Partial list of continuous functions and the values of x for which they are continuous.
1. Polynomials for all x.
2. Rational function, except for x’s that give
division by zero.
3.
n
x
(n odd) for all x.
4.
n
x
(n even) for all 0x
³
.
5.
x
e for all x.
6. ln x for 0x
>
.
7.
(
)
cos
x
and
(
)
sin
x
for all x.
8.
(
)
tan
x
and
(
)
sec
x
provided
33
,,,,,
2222
x
pppp
¹--LL
9.
(
)
cot
x
and
(
)
csc
x
provided
,2,,0,,2,x
pppp
¹--
LL
Intermediate Value Theorem
Suppose that
(
)
fx is continuous on [a, b] and let M be any number between
(
)
fa and
(
)
fb.
Then there exists a number c such that acb
<<
and
(
)
fcM=.
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