Calculus Limits Cheat Sheet - Pauls Online Math Notes

Calculus Cheat Sheet
Limits
Definitions
Precise Definition : We say lim f
x a
( x)
= L if
Æ
for every e > 0 there is a d > 0 such that
whenever 0 x a d f x - L < e .
< - < then ( )
“Working” Definition : We say lim ( )
f x = L
x a
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 ( )
+ xÆa
Æ
f x = L. This has
the same definition as the limit except it
requires x> a.
Left hand limit : lim ( )
- xÆa
f x = L. This has the
Limit at Infinity : We say lim ( )
same definition as the limit except it requires negative.
x< a.
Relationship between the limit and one-sided limits
lim f
x a
( x)
= L fi lim f ( x) = lim f ( x)
= L lim f ( x) lim f ( x)
L
+ -
+ -
Æ
xÆa xÆa
lim
( ) ≠ lim ( ) fi lim ( )
f x f x
+ -
xÆa xÆa
Assume lim ( ) and lim ( )
f x
x Æ a
x Æ
a
g x
1. limÈcf ( x) ˘ = clim
f ( x)
Î ˚
xÆa xÆa
2. limÈf ( x) ± g( x) ˘ = lim f ( x) ± lim g( x)
Î ˚
xÆa xÆa xÆa
3. limÈf ( x) g( x) ˘ = lim f ( x) lim g( x)
Î ˚
xÆa xÆa xÆa
Note : sgn( a ) = 1 if a > 0 and ( )
1. lim x
x
e =• & lim e = 0
xƕ
xÆ-•
2. limln ( x)
=• & lim ln ( x)
xƕ
xÆ0
+
xƕ
f x = L 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 ( )
xÆ-•
except we require x large and negative.
Infinite Limit : We say lim ( )
f x
x Æ a
f x = L
=• 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 ( )
f x
x Æ a
except we make f ( x ) arbitrarily large and
xÆa xÆa
f x
x Æ a
= = fi lim ( )
Does Not Exist
Properties
both exist and c is any number then,
4.
( x)
( )
È f ˘ lim f
xÆa
lim Í ˙ =
xÆa
Îg x ˚ lim g x
xÆa
n
5. limÈf ( x) ˘ lim f ( x)
= È ˘
Î ˚ Î ˚
xÆa xÆa
6. limÈn
f ( x) ˘ = n lim f ( x)
Î
xÆa xÆa
Basic Limit Evaluations at ±•
sgn a =- 1 if a < 0 .
=-•
b
3. If r > 0 then lim
r
= 0
xƕ
x
r
4. If r > 0 and x is real for negative x
b
then lim
r
= 0
xÆ-•
x
5. n even : lim x
n =•
˚
xƱ•
6. n odd : lim x
n =• & lim
xƕ
=-•
f x = L
x a
Æ
( x)
provided ( )
( )
g x
xÆa n
xÆ-•
x
n
lim ≠ 0
=-•
n
7. n even : lim ax L bx c sgn ( a)
xƱ•
+ + + = •
n
8. n odd : limax L bx c sgn ( a)
xƕ
+ + + = •
n
9. n odd : lim ax L cx d sgn ( a)
xÆ-•
+ + + =- •
Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.
© 2005 Paul Dawkins

Calculus Cheat Sheet
Continuous Functions
If f ( x)
is continuous at a then ( ) ( )
Continuous Functions and Composition
f ( x ) is continuous at b and lim g
x a
( x)
= b then
Æ
( ) ( )
( ( )) lim ( )
lim f g x = f g x = f b
xÆa xÆa
Factor and Cancel
2
x + 4x-12
x- 2 x+
6
lim
= lim
xÆ2 2
x -2x xÆ2
x x-2
( )( )
( )
x+
6 8
= lim = = 4
xÆ2
x 2
Rationalize Numerator/Denominator
3- x 3- x 3+
x
lim = lim
xÆ9 2
9
2
x -81 xÆ
x - 81 3 + x
9-x
-1
= lim
= lim
xÆ9 2
xÆ9
x - 81 3+ x x+ 9 3+
x
( )( ) ( )( )
-1 1
= =-
18 6 108
( )( )
Combine Rational Expressions
1Ê
1 1ˆ
1Ê
x- ( x+
h)
ˆ
lim Á - ˜= lim
hÆ0 h x h x hÆ0
Ë + ¯ hÁ x( x h)
˜
Ë + ¯
1Ê
-h
ˆ -1 1
= lim
lim
hÆ0 2
hÁ
x( x h) ˜
= =-
hÆ0
Ë + ¯ x( x+
h)
x
Evaluation Techniques
L’Hospital’s Rule
lim f x = f a
f
x Æ a
( x)
0 f x
If lim = or lim
xÆa
g x 0 x Æ a g x
( )
( )
=
( )
( )
( )
f x f¢
x
lim lim
xÆa g x xÆa
g¢
x
( )
( )
±•
= ±•
then,
a is a number, • or -•
Polynomials at Infinity
q x are polynomials. To compute
p( x ) and ( )
p( x)
lim
xƱ•
q( x)
of both p( x ) and ( )
factor largest power of x in q( x)
out
q x then compute limit.
( )
2 4 4
2
3x
-4 x 3 -
2 3 -
x
x2
3
lim = lim = lim
x
2 2
5
=-
Æ-• 5x-2x xÆ-• x
xÆ-•
x
-2 2
5
( x
-2)
Piecewise Function
2
Ï x + 5 if x 0 .
x ≠L, -2 p, -p,0, p,2 p,
L
Intermediate Value Theorem
Suppose that f ( x ) is continuous on [a, b] and let M be any number between f ( a ) and ( )
Then there exists a number c such that a< c< b and f ( c)
= M .
f b .
Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.
© 2005 Paul Dawkins