Transcript 2.3 - Calculating Limits Using The Limit Laws
2.3 - Calculating Limits Using The Limit Laws
1
Basic Limit Laws
1. lim
x
a c
c
(
a
,
c
)
| a y
=
c
2. lim
x
a x
a
(
a
,
a
)
| a y
=
x a x
a x n
a n
where
n
is a positive integer.
b x
a n x
n a
where
n
is a positive integer.
2
Limit Laws Generalized
Suppose that
c
is a constant and the following limits exist
x
a f x x
a g x
1 .
x
lim
a
f
(
x
)
g
(
x
)
x
lim
a f
(
x
)
x
lim
a g
(
x
) 2 .
x
lim
a
f
(
x
)
g
(
x
)
x
lim
a f
(
x
)
x
lim
a g
(
x
) 3 .
x
lim
a
cf
(
x
)
c x
lim
a f
(
x
) 3
Limit Laws Generalized
4 .
x
lim
a
f
(
x
)
g
(
x
)
x
lim
a f
(
x
)
x
lim
a g
(
x
) 5 .
x
lim
a
f g
( (
x
)
x
)
x
lim
a x
lim
a f
(
x
)
g
(
x
)
a x
a n
x
lim
a
n
where
n
is a positive integer.
b x
a n
n x
lim ( )
a
where
n
is a positive integer.
4
Examples
Evaluate the following limits. Justify each step using the laws of limits.
1 .
x
lim 3 3
x
2 2
x
5 2 .
lim
x
1 3
x
2
x
5 3 .
lim
x
2 3
x
2 2
x
5
Direct Substitution Property
If
f
is a
polynomial
or a
rational function
and
a
is in the domain of
f
, then
x
lim
a f
(
x
)
f
(
a
) 6
Examples
You may encounter limit problems that seem to be impossible to compute or they appear to not exist. Here are some tricks to help you evaluate these limits
.
1. If
f
is a
rational function
or complex: a. Simplify the function; eliminate common factors.
b. Find a common denominator.
c. Perform long division.
2. If a
root function
exists, rationalize the numerator or denominator. 3. If an
absolute values function
exists, use one-sided limits and the definition.
a
a if a a if a
0 0 7
Direct Substitution Property
Evaluate the following limits, if they exist.
1 .
lim
x
1
x
3
x
2 1 1 2 .
h
lim 0 1
h
1
h
3 .
t
lim 0
t
1
t
2 1 1 4 .
lim
x
2
x
2
x
2 8
Theorem
If
f
(
x
)
g
(
x
) when
x
is near
a
(except possibly at
a
) and the limits of
f
and
g
both exist as
x
approaches
a
, then
x
lim
a f
(
x
)
x
lim
a g
(
x
) 9
The Squeeze (Sandwich) Theorem
If
f
(
x
)
g
(
x
) possibly at
a
) and
h
(
x
) when
x
is near
a
(except
x
lim
a f
(
x
)
x
lim
a h
(
x
)
L
then
x
lim
a g
(
x
)
L
10
Example
x
lim 0
x e
sin( /
x
) 0
Strategy
To begin, bind a part of the function (usually the trigonometric part if present) between two real numbers. Then create the original function in the middle.
11
You Try It
Evaluate the following limits, if they exist, in groups of no more than three members.
1. lim
x
1 10
x x
1 2. lim
x
2 1
x
2
x
2 2 2
x
3. lim
t
7
t
7 7
t
4. lim
x
0 2
x x
1 12