2.3
Calculating Limits Using the limit Laws
Limit
Laws
Suppose that c is a constant
and the limits 
and 
exist, then
1. 
The limit of a sum is the sum of the
limits.
2. 
The limit of a difference is the
difference of the limits.
3. 
The limit of a constant times a
function is the constant times the limit of the function.
4. 
The limit of a product is the product
of the limits.
5. 
if 
The limit of a quotient is the
quotient of the limits – provided that limit of the denominator is not 0.
6. 
Example:
![[image]](Section%202.3_files/image019.gif)
Use the
Limit Laws and the graphs of f and g above to evaluate the following
limits, if they exist
Example:
Assume 
, 
and 
, find the limits that exist and indicate which rule
justifies the limit.
More
Specialized Limits:
7.
8. 
9.
where
n is a positive integer
10.
where
n is a positive integer
(If n is
odd, we assume, we assume that a > 0)
11.
where
n is a positive integer
(If n
is even, we assume, we assume that 
> 0)
Examples:
Evaluate the limit and justify each
step by indicating the appropriate Limit Law(s).
Consider the limit of a polynomial,
for example, 
Similarly, consider the limit of a
rational function, for example, 
Direct Substitution Property
If f is a polynomial or a rational function
and a is in the domain of f, then
Note:
a must be in the domain of f!!
Sometimes, algebra is necessary
before evaluating f at a.
If 
when
, then 
, provided the limits exist.
Right and Left Handed Limits
Theorem
PieceWise Function 
at 0, 1, and 2
Theorem
If 
when x is near a (except possibly at a) and the limits of f and g both exist as x
approaches a, then 
The Squeeze Theorem
(Sometimes called the Sandwich Theorem)
If 
when x is near a (except possibly at a)
and
Example
#37 Use the Squeeze
Theorem to prove that 
Assignment: 2.3 page 84; 1-4, 5-9 odd, 11-27 odd