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Calculus Cheat Sheet
Limits
Definitions
Precise Definition : We say lim fxL= if Limit at Infinity : We say lim fxL= if we
xa® ( ) x®¥ ( )
for every e > 0 there is a d > 0 such that can make fx as close to L as we want by
( )
whenever 0 .
There is a similar definition for lim fx=-¥
xa® ( )
Left hand limit : lim fxL= . This has the
- ( ) except we make fx arbitrarily large and
xa® ( )
same definition as the limit except it requires negative.
xa< .
Relationship between the limit and one-sided limits
lim fxL= Þ limfx==lim fxL limfx==lim fxL Þ lim fxL=
( ) ( ) ( ) ( ) ( ) ( )
xa® +- +-xa®
x®®axa x®®axa
limfx¹lim fx Þ lim fx Does Not Exist
( ) ( ) ( )
+-
x®®axa xa®
Properties
Assume lim fx and limgx both exist and c is any number then,
( ) ( )
xa® xa®
lim fx
1. liméùcfx=clim fx éù()
( ) ( ) fx
ëû () xa®
x®®axa 4. lim = provided lim0gx¹
êú ( )
xa® gxlimgx xa®
() ()
ëû
xa®
2. liméùfx±gx=±limfxlimgx n
( ) ( ) ( ) ( ) n
ëû
x®ax®®axa éù
5. liméùfx= lim fx
() ()
ëû
x®®axa
ëû
n
éù
n
6. limfx=limfx
3. liméùfxgx=limfxlimgx () ()
( ) ( ) ( ) ( )
ëû ëû
x®ax®®axa x®®axa
Basic Limit Evaluations at ±¥
Note : sgn1a = if a > 0 and sgn1a =- if a < 0.
( ) ( )
x x n
1. lime =¥ & lim0e = 5. n even : lim x =¥
x®¥ x®-¥ x®±¥
2. limln(x) =¥ & limln(x) =-¥ 6. n odd : lim xn =¥ & lim xn =-¥
x®¥ x®0- x®¥ x®-¥
b n
7. n even : limax+L+bx+ca=¥sgn
3. If r > 0then lim0= x®±¥ ( )
x®¥ xr
n
r 8. n odd : limax+L+bx+ca=¥sgn
4. If r > 0 and x is real for negative x x®¥ ( )
b n
9. n odd : limax+L+cx+da=-¥sgn
then lim0= x®-¥ ( )
x®-¥xr
Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins
Calculus Cheat Sheet
Evaluation Techniques
Continuous Functions L’Hospital’s Rule
If fxis continuous at a thenlim fx= fa fx fx
( ) xa® ( ) ( ) If lim ( ) = 0 or lim ( ) = ±¥ then,
xa® xa®
gx 0 gx ±¥
() ()
Continuous Functions and Composition ¢
fxfx
( ) ( )
fx is continuous at b and limgxb= then lim=lim a is a number, ¥ or -¥
( ) ( ) x®®axa
¢
gxgx
xa® () ()
limfgx==flimgxfb Polynomials at Infinity
() () ()
( ) ( )
x®®axa px and qx are polynomials. To compute
( ) ( )
Factor and Cancel px
2 xx-+26 lim ( ) factor largest power of x out of both
xx+-412 ( )( )
lim=lim x®±¥ qx
2 ()
xx®®22
x--22xxx
( ) px and qx and then compute limit.
x+68 ( ) ( )
=lim4== 2 4 4
x®2 2 x3-
x 2 ( 2 ) 3- 2
3x-43
x
x
Rationalize Numerator/Denominator lim=lim=lim =-
2 2 5 5
x®-¥xx®-¥®-¥
5xx--222
x-2
( ) x
3-x33-+xx x
lim=lim Piecewise Function
22
xx®®99
xx--8181
3+x 2
ì
91--x xx+5if 2<-
lim gx where gx=
==limlim ( ) ()í
2 x®-2 1-3xxif 2³-
xx®®99 î
x-813+xxx++93
( )( ) ( )( ) Compute two one sided limits,
-11 2
limgxx=lim+=59
==- ( )
--
186108 xx®-22®-
( )() limgxx=lim1-=37
( )
++
Combine Rational Expressions xx®-22®-
One sided limits are different so lim gx
æö ( )
x-+xh
( )
1111
æö x®-2
lim-=lim ç÷
ç÷
ç÷
hh®®00
hx++hxhxxh
èø( ) doesn’t exist. If the two one sided limits had
èø
been equal then lim gx would have existed
æö ( )
1--h 11 x®-2
=lim=lim =-
ç÷ 2
ç÷ and had the same value.
hh®®00
hxx++hxxhx
( ) ( )
èø
Some Continuous Functions
Partial list of continuous functions and the values of x for which they are continuous.
1. Polynomials for all x. 7. cos(x) and sin(x) for all x.
2. Rational function, except for x’s that give 8. tan(x) and sec(x) provided
division by zero.
3. n x (n odd) for all x. 33pppp
x ¹LL,--,,,,
4. n x (n even) for all x ³ 0 . 2222
5. ex for all x. 9. cot(x) and csc(x) provided
6. ln x for x > 0 . x ¹LL,--2p,p,0,pp,2,
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 a<,0
( ) ()
( ) 2 ( )
dx dx 1-x dxx
d 2 d -1 1 d 1
tanxx=sec lnxx=¹,0
( ) ( cos x) =- ( )
dx dx 1-x2 dxx
d d 1 d 1
secx=secxxtan -1 logxx=>,0
( ) ()
tan x = ( a )
dx ( ) 2 dxxaln
dxx1+
Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins
Calculus Cheat Sheet
Chain Rule Variants
The chain rule applied to some specific functions.
nn-1
d ¢ d ¢
1. éfxù=néùfxfx 5. coséfxù=-fxsinéùfx
() () () () () ()
( ) ( )
dx ëûëû dx ëûëû
dfxfx d
() ¢ () ¢ 2
2. ee=fx 6. tanéfxù=fxsec éùfx
() () () ()
( ) ( )
dx dx ëûëû
¢ d
fx
d () ¢
7. secf(x)=f(x)secf(x)tan fx()
3. ln éùfx = ( [ ]) [ ] [ ]
()
( )
ëû dx
dxfx
()
¢
fx
d d -1 ()
8. tan éùfx =
()
¢ ( ) 2
4. sinéfxù=fxcoséùfx ëû
() () ()
( )
ëûëû dx 1+éùfx
dx ()
ëû
Higher Order Derivatives
th
The Second Derivative is denoted as The n Derivative is denoted as
2 n
2 df n df
¢¢ () ()
fx==fx and is defined as fx= and is defined as
() () dx2 () dxn
¢ ¢
nn-1
¢¢¢ () ()
fx=fx, i.e. the derivative of the
() ()
( ) fx=fx, i.e. the derivative of
() ()
( )
¢ n-1
first derivative, fx. st ()
() the (n-1) derivative, fx.
()
Implicit Differentiation
¢ 2xy-932
Find y if e+xy=+sinyx11 . Remembery=yx here, so products/quotients of x and y
() ()
will use the product/quotient rule and derivatives of y will use the chain rule. The “trick” is to
¢
differentiate as normal and every time you differentiate a y you tack on a y (from the chain rule).
¢
After differentiating solve for y.
2xy-9223
¢¢¢
e2-9y+3xy+2xyy=+cosyy11
( ) ()
2xy-922
2x--9y2xy9223 11--23exy
¢¢¢¢
2ee-9y+3xy+2xyy=cosyyy+11 Þ=
() 329xy-
2xyy--9ecos
32x--9y2xy922 ()
¢
2xy-9ee-cosyy=11--23xy
()
( )
Increasing/Decreasing – Concave Up/Concave Down
Critical Points
xc= is a critical point of fx provided either Concave Up/Concave Down
()
¢¢
1. If fx>0 for all x in an interval I then
¢ ¢ ()
1. fc=0 or 2. fc doesn’t exist.
() ()
fx is concave up on the interval I.
()
Increasing/Decreasing ¢¢
2. If fx<0 for all x in an interval I then
()
¢
1. If fx>0 for all x in an interval I then
() fx is concave down on the interval I.
()
fx is increasing on the interval I.
()
¢ Inflection Points
2. If fx<0 for all x in an interval I then
()
xc= is a inflection point of fx if the
fx is decreasing on the interval I. ()
()
¢ concavity changes at xc=.
3. If fx=0 for all x in an interval I then
()
fx is constant on the interval I.
()
Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins
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