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
fx L
→
=
if
for every
0
ε
>
there is a
0
δ
>
such that
whenever
0
<−<
then
−<
.
“Working” Definition :
We say
lim
→
=
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
x
a
=
.
Right hand limit :
lim
+
→
=
. This has
the same definition as the limit except it
requires
x
a
>
.
Left hand limit :
lim
−
→
=
. This has the
same definition as the limit except it requires
x
a
<
.
Limit at Infinity :
We say
(
)
lim
x
→∞
=
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
→−∞
=
except we require
x
large and negative.
Infinite Limit :
We say
(
)
lim
→
= ∞
if we
can make
(
)
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
→
= −∞
except we make
(
)
arbitrarily large and
negative.
Relationship between the limit and onesided limits
lim
→
=
⇒
lim
lim
+−
→→
==
(
)
lim
lim
=
=
⇒
(
)
lim
→
=
lim
lim
≠
⇒
lim
→
Does Not Exist
Properties
Assume
lim
→
and
lim
gx
→
both exist and
c
is any number then,
1.
lim
lim
cf x
c
f x
=
⎡⎤
⎣⎦
2.
lim
lim
lim
fx gx
→
±=
±
3.
() ()
lim
lim
lim
fxgx
→
=
4.
(
)
lim
lim
lim
g
xg
x
→
→
→
=
⎢⎥
provided
(
)
lim
0
→
≠
5.
lim
lim
n
n
⎡
⎤
=
⎣
⎦
6.
lim
lim
n
n
=
Basic Limit Evaluations at
±∞
Note :
sgn
1
a
=
if
0
a
>
and
sgn
1
a
=−
if
0
a
<
.
1.
lim
x
x
→∞
=∞
e
&
lim
0
x
x
→− ∞
=
e
2.
lim ln
x
x
→∞
&
0
lim ln
x
x
−
→
=−∞
3.
If
0
r
>
then
lim
0
r
x
b
x
→∞
=
4.
If
0
r
>
and
r
x
is real for negative
x
then
lim
0
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 :
(
)
lim
sgn
n
x
ax
bx c
a
+
++
=
∞
"
8.
n
odd :
(
)
lim
sgn
n
x
a
→∞
+
=
∞
"
9.
n
odd :
(
)
lim
sgn
n
x
cx d
a
→−∞
+
++=
−
∞
"
Calculus Cheat Sheet
Visit
http://tutorial.math.lamar.edu
for a complete set of Calculus notes.
©
2005 Paul Dawkins
Evaluation Techniques
Continuous Functions
If
(
)
is continuous at
a
then
(
)
(
)
lim
fx fa
→
=
Continuous Functions and Composition
(
)
is continuous at
b
and
(
)
lim
gx b
→
=
then
lim
lim
fgx
f
fb
Factor and Cancel
(
)
2
2
22
2
26
41
2
lim
lim
68
lim
4
2
xx
x
x
x
x
x
→
−+
=
−−
+
=
Rationalize Numerator/Denominator
(
)
99
2
33
3
lim
lim
81
81 3
91
lim
lim
81 3
9 3
11
18 6
108
x
x
x
x
x
+
=
+
+
+
−
−
Combine Rational Expressions
(
)
00
2
11 1
1
lim
lim
1
lim
lim
hh
xxh
hxh x
h xxh
h
hxxh
x
⎛⎞
−=
⎜⎟
⎝⎠
=
−
L’Hospital’s Rule
If
(
)
0
lim
0
→
=
or
(
)
lim
→
=
then,
(
)
(
)
lim
lim
f x
g x
′
=
′
a
is a number,
∞
or
−∞
Polynomials at Infinity
(
)
px
and
(
)
qx
are polynomials.
To compute
(
)
lim
x
factor largest power of
x
out of both
(
)
and
(
)
and then compute limit.
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 Spring '08
 Skrypka
 Calculus, Derivative, Limits, lim, dx, Paul Dawkins

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