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 ( )limx a

f x L→

= if

for every 0ε > there is a 0δ > such that whenever 0 x a δ< − < then ( )f x L ε− < . “Working” Definition : We say ( )lim

x af 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= . Right hand limit : ( )lim

x af x L

+→= . This has

the same definition as the limit except it requires x a> . Left hand limit : ( )lim

x af x L

−→= . This has the

same definition as the limit except it requires x a< .

Limit at Infinity : We say ( )limx

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

xf x L

→−∞=

except we require x large and negative. Infinite Limit : We say ( )lim

x af x

→= ∞ 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 af x

→= −∞

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

Relationship between the limit and one-sided limits ( )lim

x af x L

→= ⇒ ( ) ( )lim lim

x a x af x f x L

+ −→ →= = ( ) ( )lim lim

x a x af x f x L

+ −→ →= = ⇒ ( )lim

x af x L

→=

( ) ( )lim limx a x a

f x f x+ −→ →

≠ ⇒ ( )limx a

f x→

Does Not Exist

Properties

Assume ( )limx a

f x→

and ( )limx a

g x→

both exist and c is any number then,

1. ( ) ( )lim limx a x a

cf x c f x→ →

=

2. ( ) ( ) ( ) ( )lim lim lim

x a x a x af x g x f x g x

→ → →± = ±

3. ( ) ( ) ( ) ( )lim lim lim

x a x a x af x g x f x g x

→ → →=

4. ( )( )

( )( )

limlim

limx a

x ax a

f xf xg x g x

→

→→

=

provided ( )lim 0

x ag x

→≠

5. ( ) ( )lim limnn

x a x af x f x

→ → =

6. ( ) ( )lim limn nx a x a

f x f x→ →

=

Basic Limit Evaluations at ± ∞ Note : ( )sgn 1a = if 0a > and ( )sgn 1a = − if 0a < .

1. lim xx→∞

= ∞e & lim 0xx→− ∞

=e

2. ( )lim lnx

x→∞

= ∞ & ( )0

lim lnx

x+→

= −∞

3. If 0r > then lim 0rx

bx→∞

=

4. If 0r > and rx is real for negative x

then lim 0rx

bx→−∞

=

5. n even : lim n

xx

→±∞= ∞

6. n odd : lim n

xx

→∞= ∞ & lim n

xx

→− ∞= −∞

7. n even : ( )lim sgnn

xa x b x c a

→±∞+ + + = ∞L

8. n odd : ( )lim sgnn

xa x b x c a

→∞+ + + = ∞L

9. n odd : ( )lim sgnn

xa x c x d a

→−∞+ + + = − ∞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 af x f a

→=

Continuous Functions and Composition

( )f x is continuous at b and ( )limx a

g x b→

= then

( )( ) ( )( ) ( )lim limx a x a

f g x f g x f b→ →

= =

Factor and Cancel ( ) ( )

( )2

22 2

2

2 64 12lim lim2 2

6 8lim 42

x x

x

x xx xx x x x

xx

→ →

→

− ++ −=

− −

+= = =

Rationalize Numerator/Denominator

( )( ) ( ) ( )

( ) ( )

2 29 9

29 9

3 3 3lim lim81 81 3

9 1lim lim81 3 9 3

1 118 6 108

x x

x x

x x xx x x

xx x x x

→ →

→ →

− − +=

− − +− −

= =− + + +

−= = −

Combine Rational Expressions ( )

( )

( ) ( )

0 0

20 0

1 1 1 1lim lim

1 1 1lim lim

h h

h h

x x hh x h x h x x h

hh x x h x x h x

→ →

→ →

− + − = + + − −

= = = − + +

L’Hospital’s Rule

If ( )( )

0lim0x a

f xg x→

= or ( )( )

limx a

f xg x→

± ∞=

± ∞ then,

( )( )

( )( )

lim limx a x a

f x f xg x g x→ →

′=

′ a is a number, ∞ or −∞

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

( )( )

limx

p xq x→±∞

factor largest power of x in ( )q x out

of both ( )p x and ( )q x then compute limit.

( )( )

22

2 2

2 24 4

55

3 33 4 3lim lim lim5 2 2 22x x x

xx

x xxx

x x x→−∞ →−∞ →−∞

− −−= = = −

− −−

Piecewise Function

( )2

limx

g x→−

where ( )2 5 if 2

1 3 if 2x x

g xx x

+ < −=

− ≥ −

Compute two one sided limits, ( ) 2

2 2lim lim 5 9

x xg x x

− −→− →−= + =

( )2 2

lim lim 1 3 7x x

g x x+ +→− →−

= − =

One sided limits are different so ( )2

limx

g x→−

doesn’t exist. If the two one sided limits had been equal then ( )

2limx

g x→−

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 ≥ . 5. xe for all x. 6. ln x for 0x > .

7. ( )cos x and ( )sin x for all x.

8. ( )tan x and ( )sec x provided 3 3, , , , ,2 2 2 2

x π π π π≠ − −L L

9. ( )cot x and ( )csc x provided , 2 , ,0, , 2 ,x π π π π≠ − −L L

Intermediate Value Theorem

Suppose that ( )f x is continuous on [a, b] and let M be any number between ( )f a and ( )f b .

Then there exists a number c such that a c b< < and ( )f c M= .

CH.1 LIMITS WS #1 NAME PER.

LIMITS AT NON-BREAKING POINTS

1) limx→1

x3 + 2x − 5 = 2) limx→9

x2 − 9x − 3

= 3) limx→−27

x3 =

4) limx→3

−1x

= 5) limx→2

x + 2 = 6) limx→1

x −1x +1

=

HOLES IN THE GRAPH

7) limx→2

x2 + 3x −10x − 2

= 8) limx→1+

x2 −1x −1

= 9) limx→4

x2 −164 − x

=

10) limx→−2

x +11 − 3x + 2

= 11) limx→64

8 − xx − 64

= 12) limx→7

x2 − 49x +18 − 5

=

RADICALS

13) lim

x→3− 3− x = 14) lim

x→−5− x + 5 = 15) lim

x→4+ x2 −16 +1=

16) limx→5+

25 − x2 = 17) limx→5−

25 − x2 = 18) limx→5

25 − x2 =

ASYMPTOTES

19) limx→2+

3x − 2

= 20) limx→2−

−52 − x

= 21) limx→−8

xx + 8

=

22) limx→−3+

3xx + 3

= 23) limx→6

−8x − 6( )2 = 24) lim

x→π2+ tan x =

TRIG. FUNCTIONS

25) limx→0

sin xx

= 26) limx→0

cos xx

= 27) limx→0

tan xx

= 28) limx→0

sin x tan xx2 =

29) limx→0

3sin 3x8x

= 30) limx→0

6sin x cos x5x

= 31) limx→π

2 5sin 3x

x=

PIECE-WISE FUNCTIONS

f x( ) =3− x x < −32x +1 −3 ≤ x < 4

9 x ≥ 4

⎧⎨⎪

⎩⎪

32) limx→−5

f x( ) = 33) limx→−3+

f x( ) = 34) limx→−3−

f x( ) = 35) limx→−3

f x( ) =

36) limx→4+

f x( ) = 37) limx→4−

f x( ) = 38) limx→4

f x( ) = 39) limx→7+

f x( ) =

LIMITS THAT APPROACH INFINITY

40) limx→∞

3− 7x2

14x2 +1= 41) lim

x→∞ x

3 + 5−x2 = 42) lim

x→∞ 3− 2xx2 + 9

= 43) limx→−∞

4x − 57x +1

=

44) limx→∞

3− xx − 3

= 45) limx→∞

5 = 46) limx→∞

25 − x

= 47) limx→−∞

7x5 + 33x2 + 5

=

Find the vertical asymptotes and holes for each.

48) f x( ) = 3x − 25 − x

49) f x( ) = x2 + 2x −15x − 3

50) f x( ) = 8 − x8 − x

vert.asym. hole vert.asym. hole vert.asym. hole

51) f x( ) = 8x −13x2 + 5

52) f x( ) = x + 2( ) x − 3( ) x + 4( )x + 2( ) x + 4( ) x + 7( ) 53) f x( ) = 2x +1( ) x + 3( )

x − 4( ) 2x +1( )vert.asym. hole vert.asym. hole vert.asym. hole

Write an equation that meets the requirements.

54) Hole at : x = 3 55) Hole at : 2, 5( ) 56) vert. asym. : x = − 4

vert. asym. : x = −5 Hole at −2, −5 2

⎛⎝⎜

⎞⎠⎟

57) limx→0

f x( ) = 12

vert. asym. : x = − 7 2

58) limx→∞

f x( ) = 3 vert. asym. : x = −2

59) lim

x→−2+f x( ) = 60) lim

x→−2−f x( ) = 61) lim

x→0f x( ) = 62) lim

x→2f x( ) =

63) limx→4

f x( ) = 64) limx→6

f x( ) = 65) limx→−7−

f x( ) = 66) limx→∞

f x( ) =

67) limx→−∞

f x( ) = 68) limx→−5

f x( ) = 69) limx→−7

f x( ) = 70) limx→−2

f x( ) =

71) limx→2+

f x( ) = 72) limx→2−

f x( ) = 73) limx→1

f x( ) = 74) limx→−1

f x( ) =

75) limx→−7+

f x( ) = 76) limx→−5−

f x( ) = 77) limx→−5+

f x( ) = 78) limx→4+

f x( ) =

79) The graph is discontinuous at which points:

80) limx→1+

x −1x −1

= 81) limx→2+

x − 2( )2

x − 2=

9-9 -8 87-7

-5

-4 -3 -2 -1-6 -5

-4-3-2-1

54

321 654

321

0

CALCULUS AB / BC CH.1 LIMITS WS #2 NAME PER.

1) lim

x→7 x2 + 2 = 2) lim

x→4 x

2 −16x − 4

= 3) limx→−27

6x + 27

=

4) limx→−5

x3 +125x + 5

= 5) limx→1+

−9x −1

= 6) limx→0

2sin x cos x tan xx2 =

7) limx→∞

3x − 8x + 2

= 8) limx→100

10 − xx −100

= 9) limx→7

7 − x =

10) limx→15−

15 − x = 11) limx→8

x + 5 = 12) limx→4+

-xx − 4

=

13) limx→0

sin2 x tan xx2 = 14) lim

x→∞ x7

x5 − 9= 15) lim

x→12 25 − x2 =

16) limx→7−

15x − 7

= 17) limx→0

cos xx2 = 18) lim

x→π cos xx2 =

19) limx→41

41− x = 20) limx→6

x − 6x − 6( )2 = 21) lim

x→0 tan9x

57x=

22) limx→0

sin x3x

= 23) limx→0−

cos xx

= 24) limx→0

tan8x32x

=

25) limx→0+

sin2 x tan2 xx5 = 26) lim

x→π4 sin x cos x

4x= 27) lim

x→∞

sin xx

=

f x( ) =x2 x < −3

3− 2x −3 ≤ x < 41 x ≥ 4

⎧

⎨⎪

⎩⎪

28) limx→−5

f x( ) = 29) limx→−3+

f x( ) = 30) limx→−3−

f x( ) = 31) limx→−3

f x( ) =

32) limx→4+

f x( ) = 33) limx→4−

f x( ) = 34) limx→4

f x( ) = 35) limx→7+

f x( ) =

Find the vertical asymptotes and holes for each.

36) f x( ) = 3− xx − 3

37) f x( ) = x +1( ) x − 3( )x +1( ) x +10( ) x − 3( ) 38) f x( ) = 3x + 5( ) x + 6( )

x − 7( ) 4x + 5( )vert.asym. hole vert.asym. hole vert.asym. hole

Write an equation that meets the requirements.

39) Hole at : x = 8 40) limx→0

f x( ) = 29

41) Hole at : − 4, 2( ) 42) v. asym. : x = 2

v. asym. : x = 7 v. asym. : x = −94

Hole at 6, 3( )

43) lim

x→−3 −f x( ) = 44) lim

x→3 +f x( ) = 45) lim

x→−2f x( ) = 46) lim

x→2f x( ) =

47) limx→4

f x( ) = 48) limx→4+

f x( ) = 49) limx→3 −

f x( ) = 50) limx→∞

f x( ) =

51) limx→−∞

f x( ) = 52) limx→− 3 +

f x( ) = 53) limx→3

f x( ) = 54) limx→0

f x( ) =

-5

-4

-3 -2-1

-5

-4

-3-2

-1 5

4

3

21

5

4

3

210

The 7 BIG

Theorems of C

alculus

Intermediate V

alue Theorem

Let f be a continuous function on [a, b]. Then there exists a num

ber c such that a < c < b w

here f (c) = k on [f (a), f (b)]

If differentiable, then continuous!

Mean V

alue Theorem

for Derivatives

Let f be a function such that 1) f is continuous on [a, b], and 2) f is differentiable on (a, b). Then there exists a num

ber c (a < c < b) such that

()

()

'()

fb

fa

fc

ba

Rolle’s T

heorem

Let f be a function such that 1) f is continuous on [a, b], 2) f is differentiable on (a, b), and 3) f (a)=f (b). Then there exists a num

ber c (a < c < b) such that '(

)0

fc

Extrem

e Value T

heorem

Every continuous function has an absolute m

aximum

and an absolute minim

um on a

closed interval.

Mean V

alue Theorem

for Integrals

If f is continuous on [a, b], then there exists a num

ber c in the closed interval [a, b] such that

1(

)(

)ba

fc

fx

dxb

a

The Fundam

ental Theorem

of Integral Calculus

Let f be a continuous function on [a, b].

Part I: If (

)(

)xa

Gx

ft

dt

for every x in [a, b], then G (x) is an antiderivative of f on [a, b],

'()

()

Gx

fx

.

Part II: If F(x) is an antiderivative of f(x) on [a, b], then (

)(

)(

).ba

fx

dxF

bF

a