limx cK K
The limit of a constant IS the constant.No matter what “x” approaches
2lim ( )x
f x
1lim ( )x
f x
25lim ( )x
f x
Limit Laws
lim ( ) ( ) lim ( ) lim ( )x c x c x c
f x g x f x g x
The limit of a sum is the sum of the limits.
Theorem 2.3 (1)
2
lim 5x
x
2 2
lim lim5x xx
2 5
7
lim ( ) ( ) lim ( ) lim ( )x c x c x c
f x g x f x g x
The limit of a difference is the difference of the limits.
Theorem 2.3 (2)
3
lim 7x
x
3 3
lim 7 limx x
x
7 3
4
lim ( ) lim ( )x c x ck f x k f x
The limit of a constant times a function is the constant times the limit of the function.
Theorem 2.3 (3)
2lim 6x
x 2
6 limxx
2
12
lim ( ) ( ) lim ( ) lim ( )x c x c x c
f x g x f x g x
The limit of a product is the product of the limits.
Theorem 2.3 (4)
3
limx
x x
3 3
lim limx xx x
3 3
9
lim ( )( )lim
( ) lim ( )x c
x cx c
f xf x
g x g x
The limit of a quotient is the quotient of the limits.
Theorem 2.3 (5)
6lim
3x
x
6
6
lim
lim3x
x
x
6
32
lim ( ) lim ( )nn
x c x cf x f x
Theorem 2.3 (6)
The limit of a “function raised to a power” is the “limit of the function” raised to the power.
4
2limx
x
4
2limxx
2
16
Theorem 2.3 (7)
lim ( ) lim ( )nn mm
x c x cf x f x
The limit of a “function raised to a fractional power” is the “limit of the function” raised to the fractional power.
23
8limx
x
23
8limxx
8
4
( )lim ( )
( )x c
p cr x
q c
rationalIf is funa ctionr( )
( )( )
p xr x
q x
If ( ) 0q c
2
22
5 6lim
2x
x x
x
2
2
(2) 5(2) 6
(2) 2
4
3
lim ( ) ( )x cf x f c
If is any of the six trig functionsf
for any in the domainc
lim cosx
x
cos 1
lim tanx
x
tan 0
9
3lim
9x
x
x
Sometimes when you have radicals you need to multiply by the conjugate of the numerator or denominator.
For all on ( , ),x a b
( ) ( ) ( )f x h x g x except possibly at c
a b
lim ( ) lim ( ) x c x cf x g x L
for some a c b
And
then lim ( )x ch x L
Squeeze TheoremPinching TheoremSandwich Theorem
2
0
1Find lim cos
xx
x
Notice we cannot substitute 0 for .xThere is no factoring or simplifying
There is no rationalizing.
We know that the range of the cosine function is [ 1,1]1
1 cos 1x
for all 0x
2Multiplying by .x
2
0limx
x
2
0limx
x
2
0
1lim cosxx
x