Calculus Cheat Sheet Calculus Cheat Sheet
Evaluation Techniques
Limits
Continuous Functions L Hospital s Rule
Definitions
f x f x
If f x is continuous at a then lim f x f a 0
Precise Definition : We say lim f x L if Limit at Infinity : We say lim f x L if we
x a
If lim or lim then,
x a x
x a x a
g x 0 g x
for every 0 there is a 0 such that
can make f x as close to L as we want by
Continuous Functions and Composition
f x f x
whenever 0 x a then f x L .
taking x large enough and positive.
lim lim a is a number, or
f x is continuous at b and lim g x b then
x a x a
g x g x
x a
There is a similar definition for lim f x L Polynomials at Infinity
 Working Definition : We say lim f x L
lim f g x f lim g x f b
x
x a
x ax a
p x and q x are polynomials. To compute
except we require x large and negative.
if we can make f x as close to L as we want
Factor and Cancel
p x
x 2 x 6
by taking x sufficiently close to a (on either side x2 4x 12 lim factor largest power of x in q x out
x
Infinite Limit : We say lim f x if we limlim
q x
of a) without letting x a . x 2 x 2
x a
x2 2xx x 2
of both p x and q x then compute limit.
can make f x arbitrarily large (and positive)
x 6 8
Right hand limit : lim f x L . This has lim4
4
4
by taking x sufficiently close to a (on either side
x 2
x a
x 2 x2 3
3
3x2 43
x2
x2
of a) without letting x a .
the same definition as the limit except it
limlimlim
Rationalize Numerator/Denominator
x
5x 2x2 x 5 x 5 2 2
x2 2
x
requires x a . x
3 x 3 x 3 x
lim lim
There is a similar definition for lim f x
Piecewise Function
x 9 x 9
x a
x2 81 x2 81
3 x
Left hand limit : lim f x L . This has the
x2 5 if x 2
x a except we make f x arbitrarily large and
9 x 1
lim2 g x where g x
limlim
x
same definition as the limit except it requires
1 3x if x 2
x 9 x 9
negative.
x2 81 3 x x 9 3 x
x a .
Compute two one sided limits,
Relationship between the limit and one-sided limits
1 1
lim g x lim x2 5 9
x 2 x 2
lim f x L lim f x lim f x L lim f x lim f x L lim f x L
18 6 108
x a x a
x a x a x a x a
lim g x lim1 3x 7
Combine Rational Expressions x 2 x 2
lim f x lim f x lim f x Does Not Exist
x a
x a x a
One sided limits are different so lim2 g x
x x h
1 1 1 1
x
limlim
h 0 h 0
h x h x h x x h
doesn t exist. If the two one sided limits had
Properties
been equal then lim2 g x would have existed
Assume lim f x and lim g x both exist and c is any number then,
1 h 1 1 x
x a x a
limlim
h 0 h 0
lim f x h x x h x x h x2 and had the same value.
1. lim cf x c lim f x
f x
x a
x ax a
4. lim provided lim g x 0
x a x a
g x lim g x
x a
n
2. lim f x g x lim f x lim g x Some Continuous Functions
n
x ax a x a
5. lim f x lim f x
Partial list of continuous functions and the values of x for which they are continuous.
x ax a
1. Polynomials for all x.
7. cos x and sin x for all x.
n
n
6. lim f x lim f x
3. lim f x g x lim f x lim g x
2. Rational function, except for x s that give
x ax a
x ax a x a
8. tan x and sec x provided
division by zero.
n 33
3. x (n odd) for all x.
Basic Limit Evaluations at
x , , , , ,
n 2 2 2 2
4. x (n even) for all x 0 .
Note : sgn a 1 if a 0 and sgn a 1 if a 0 .
9. cot x and csc x provided
5. ex for all x.
1. lim ex & lim ex 0 5. n even : lim xn
x x x x , 2 , ,0, , 2 ,
6. ln x for x 0 .
2. limln x & limln x 6. n odd : lim xn & lim xn
x x x
x 0
Intermediate Value Theorem
b
7. n even : lim axn bx c sgn a
3. If r 0 then lim 0 Suppose that f x is continuous on [a, b] and let M be any number between f a and f b .
x
x
xr
8. n odd : lim axn bx c sgn a
Then there exists a number c such that a c b and f c M .
4. If r 0 and xr is real for negative x
x
b
9. n odd : lim axn cx d sgn a
then lim 0
x
x
xr
Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins