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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 x Æ a f (^) ( x (^) )= L if
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 x Æ a 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.
Right hand limit : (^) x lim Æ a + f (^) ( x (^) )= L. This has
the same definition as the limit except it requires x > a.
Left hand limit : (^) x lim Æ a - f (^) ( x (^) )= L. This has the
same definition as the limit except it requires x < a.
Limit at Infinity : We say (^) x lim Æ• 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 (^) x lim Æ-• f (^) ( x (^) )= L except we require x large and negative.
Infinite Limit : We say lim x Æ a f ( 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 Æ a f (^) ( x )= -• except we make f (^) ( x ) arbitrarily large and negative. Relationship between the limit and one-sided limits lim x Æ a f (^) ( x (^) )= L fi (^) x lim Æ a + f (^) ( x (^) ) = (^) x lim Æ a - f (^) ( x ) = L x lim Æ a + f (^) ( x (^) ) = (^) x lim Æ a - f (^) ( x (^) )= L fi lim x Æ a f (^) ( x (^) )= L
x lim Æ a +^^ f^^ (^ x^ )^ π^ x lim Æ a - f^ (^ x ) fi^ lim x Æ a f^ (^ x )Does Not Exist
Properties Assume lim x Æ a f (^) ( x )and lim x Æ a g (^) ( x ) both exist and c is any number then,
- lim x Æ a ÈÎ cf (^) ( x (^) ) ˘˚ = c lim x Æ a f (^) ( x )
- lim x Æ a ÈÎ f (^) ( x (^) ) ± g (^) ( x (^) ) ˘˚= lim x Æ a f (^) ( x (^) ) ±lim x Æ ag (^) ( x )
- lim x Æ a ÈÎ f (^) ( x g ) (^) ( x (^) ) ˘˚ =lim x Æ a f (^) ( x (^) ) lim x Æ ag (^) ( x )
- (^ ) ( )
( ) ( )
lim lim lim
x a x a x a
f x^ f^ x g x g x
Æ Æ Æ
È ˘
Í ˙=
Î ˚
provided lim x Æ a g (^) ( x )π 0
- lim (^) ( ) lim (^) ( ) n n x Æ a f^ x^ x Æ a f^ x
ÈÎ ˘˚ =^ È^ ˘
Î ˚
- lim x Æ a^ ÈÎ^ n^ f (^) ( x (^) ) ˘ =˚ n lim x Æ af (^) ( x )
Basic Limit Evaluations at ± • Note : sgn (^) ( a (^) )= 1 if a > 0 and sgn (^) ( a (^) )= - 1 if a < 0.
- (^) x limÆ• e x = • & (^) x limÆ- • e x = 0
- (^) x lim ln Æ• ( x )= • & (^) x lim ln Æ 0 + ( x )= - •
- If r > 0 then lim x^ br 0 Æ• (^) x
- If r > 0 and xr is real for negative x
then lim x^ br 0 Æ-• (^) x
- n even : (^) x lim Ʊ • xn = •
- n odd : lim x Æ• xn = • & (^) x lim Æ- • xn = -•
- n even : (^) x lim Ʊ • a xn + L+ b x + c = sgn( a )•
- n odd : lim x Æ• a xn + L+ b x + c = sgn( a )•
- n odd : (^) x lim Æ-• a x n + L+ c x + d = - sgn( 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 f ( x )is continuous at a then lim x Æ a f ( x ) = f ( a )
Continuous Functions and Composition
f ( x )is continuous at b and lim x Æ a g ( x )= b then
lim x Æ a f ( g ( x )) = f ( lim x Æ ag ( x )) = f ( b )
Factor and Cancel
2 2 2 2
2
lim lim 2 2
lim 6 8 4 2
x x
x
x x^ x^ x x x x x x x
Æ Æ
Æ
+ - -^ +
Rationalize Numerator/Denominator
( )( ) (^ )^ ( )
9 2 9 2
9 2 9
lim 3 lim^3 81 81 3 lim 9 lim^1 81 3 9 3 1 1 18 6 108
x x
x x
x x x x x (^) x x x x x x
Æ Æ
Æ Æ
= -^ = -
Combine Rational Expressions
0 0
0 0 2
lim 1 1 1 lim^1
lim 1 lim^1
h h
h h
x x h h x h x h x x h h h x x h x x h x
Æ Æ
Æ Æ
Ê ˆ^ Ê^ -^ + ˆ
Á -^ ˜ =^ ÁÁ^ ˜˜
Ë +^ ¯ Ë + ¯
Ê - ˆ -
= ÁÁ ˜˜= = -
Ë +^ ¯ +
L’Hospital’s Rule
If (^ )
lim 0 x a 0
f x Æ (^) g x
= or (^ )
lim x a
f x Æ g x
then,
lim x a (^) lim x a
f x f x Æ (^) g x Æ g x
a is a number, • or -•
Polynomials at Infinity
p x ( )and q x ( )are polynomials. To compute
x^ lim
p x Ʊ • q x
factor largest power of x in q x ( )out
of both p x ( )and q x ( )then compute limit.
2 2 (^2 )
(^2 )
(^4 ) 5 5
lim lim lim x (^) 5 2 x (^) x 2 x (^) x 2 2
x^ x x x Æ-• (^) x x Æ-• (^) x Æ- •
- -^ -
Piecewise Function
x^ lim Æ- 2 g^^ (^ x )where^ (^ )^
(^2 5) if 2 1 3 if 2
x x g x x x
Ï + < -
= Ì
Ó -^ ≥ -
Compute two one sided limits,
x^ lim Æ- 2 -^^ g^^ (^ x^ )^ =^ x limÆ- 2 -^ x^2 +^5 =^9
x^ lim Æ- 2 +^^ g^^ (^ x^ )^ =^ x lim 1Æ- 2 +^ -^3 x =^7
One sided limits are different so x lim Æ- 2 g ( x )
doesn’t exist. If the two one sided limits had
been equal then lim x Æ- 2 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.
- Polynomials for all x.
- Rational function, except for x ’s that give division by zero.
- n^ x ( n odd) for all x.
- n^ x ( n even) for all x ≥ 0.
- e x for all x.
- ln x for x > 0.
7. cos ( x )and sin ( x )for all x.
8. tan ( x )and sec ( x )provided
x π L -^ p^ - p^ p^ p L
9. cot ( x )and csc ( x )provided
x π 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 f ( b ).
Then there exists a number c such that a < c < b and f ( c )= M.
