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Calculus Cheat Sheet

Limits
Definitions
Precise Definition : We say lim f ( x ) = L if
Limit at Infinity : We say lim f ( x ) = L if we
x ®a

x ®¥

for every e > 0 there is a d > 0 such that
whenever 0 < x - a < d then f ( x ) - L < e
...


“Working” Definition : We say lim f ( x ) = L

There is a similar definition for lim f ( x ) = L

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
...


x ®a

Right hand limit : lim+ f ( x ) = L
...

Left hand limit : lim- f ( x ) = L
...

There is a similar definition for lim f ( x ) = -¥
x ®a

except we make f ( x ) arbitrarily large and
negative
...

Relationship between the limit and one-sided limits
lim f ( x ) = L Þ lim+ f ( x ) = lim- f ( x ) = L
lim+ f ( x ) = lim- f ( x ) = L Þ lim f ( x ) = L
x ®a

x ®a

x ®a

x ®a

x ®a

x ®a

lim f ( x ) ¹ lim- f ( x ) Þ lim f ( x ) Does Not Exist

x ®a +

x ®a

x ®a

Properties
Assume lim f ( x ) and lim g ( x ) both exist and c is any number then,
x ®a

x ®a

1
...
lim é f ( x ) ± g ( x ) ù = lim f ( x ) ± lim g ( x )
û x®a
x ®a ë
x ®a
3
...
lim ê
provided lim g ( x ) ¹ 0
ú=
x ®a
x ®a g ( x )
ë
û lim g ( x )
x ®a
n

n
5
...
lim é n f ( x ) ù = n lim f ( x )
û
x ®a ë
x®a

Basic Limit Evaluations at ± ¥
Note : sgn ( a ) = 1 if a > 0 and sgn ( a ) = -1 if a < 0
...
lim e x = ¥ &
x®¥

2
...
If r > 0 and x r is real for negative x
b
then lim r = 0
x ®-¥ x
3
...
n even : lim x n = ¥
x ®± ¥

6
...
n even : lim a x + L + b x + c = sgn ( a ) ¥
n

x ®± ¥

8
...
n odd : lim a x n + L + c x + d = - sgn ( a ) ¥

Visit http://tutorial
...
lamar
...


x ®-¥

© 2005 Paul Dawkins

Calculus Cheat Sheet

Evaluation Techniques
Continuous Functions
L’Hospital’s Rule
f ( x) 0
f ( x) ± ¥
If f ( x ) is continuous at a then lim f ( x ) = f ( a )
x ®a
If lim
= or lim
=
then,
x ®a g ( x )
x ®a g ( x )
0
±¥
Continuous Functions and Composition
f ( x)
f ¢( x)
lim
= lim
a is a number, ¥ or -¥
f ( x ) is continuous at b and lim g ( x ) = b then
x ®a g ( x )
x ®a g ¢ ( x )

(

)

x ®a

lim f ( g ( x ) ) = f lim g ( x ) = f ( b )
x ®a

x ®a

Polynomials at Infinity
p ( x ) and q ( x ) are polynomials
...


(

(

)

Piecewise Function

)

-1
1
=(18)( 6 ) 108
Combine Rational Expressions
1æ 1
1ö
1 æ x - ( x + h) ö
lim ç
- ÷ = lim ç
÷
h ®0 h x + h
x ø h ®0 h ç x ( x + h ) ÷
è
è
ø
1 æ -h ö
1
-1
= lim ç
=- 2
÷ = lim
h ®0 h ç x ( x + h ) ÷
h®0 x ( x + h )
x
è
ø
=

)

x 2 3 - 42
3 - 42
3x 2 - 4
3
x
lim
= lim 2 5
= lim 5 x = x ®-¥ 5 x - 2 x 2
x ®-¥ x
x ®- ¥
2
x -2
x -2
ì x 2 + 5 if x < -2
lim g ( x ) where g ( x ) = í
x ®-2
î1 - 3x if x ³ -2
Compute two one sided limits,
lim- g ( x ) = lim- x 2 + 5 = 9
x ®-2

x ®-2

x ®-2+

x ®-2

lim g ( x ) = lim+ 1 - 3 x = 7

One sided limits are different so lim g ( x )
x ®-2

doesn’t exist
...


Some Continuous Functions
Partial list of continuous functions and the values of x for which they are continuous
...
Polynomials for all x
...
cos ( x ) and sin ( x ) for all x
...
Rational function, except for x’s that give
8
...

3p p p 3p
3
...

x ¹ L , - , - , , ,L
2
2 2 2
4
...

9
...
e x for all x
...
ln x for x > 0
...

Then there exists a number c such that a < c < b and f ( c ) = M
...
math
...
edu for a complete set of Calculus notes

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