Math 103: One-Sided Limits of Functions
Ryan Blair
University of Pennsylvania
Thursday September 15, 2011
Ryan Blair (U Penn)
Math 103: One-Sided Limits of Functions
Thursday September 15, 2011
1/7
Outline
1
Review
2
One-Sided Limits
Ryan Blair (U Penn)
Math 103: One-Sided Limits of Functions
Thursday September 15, 2011
2/7
Review
Definition of Limit
Definition
If f (x) is arbitrarily close to L for all x sufficiently close to
x0, we say f approaches the limit L as x approaches x0
and write:
limx→x0 f (x) = L
Ryan Blair (U Penn)
Math 103: One-Sided Limits of Functions
Thursday September 15, 2011
3/7
Review
Definition of Limit
Definition
If f (x) is arbitrarily close to L for all x sufficiently close to
x0, we say f approaches the limit L as x approaches x0
and write:
limx→x0 f (x) = L
Last time we saw
1
2
3
Limit laws
Theorems regarding polynomials and rational functions
How to evaluate a limit if there is a zero in the
denominator
Ryan Blair (U Penn)
Math 103: One-Sided Limits of Functions
Thursday September 15, 2011
3/7
Review
The Sandwich Theorem
Theorem
If f (x) ≤ g (x) ≤ h(x) when x is near c and
limx→c f (x) = limx→c h(x) = L
then limx→c g (x) = L
Ryan Blair (U Penn)
Math 103: One-Sided Limits of Functions
Thursday September 15, 2011
4/7
Review
The Sandwich Theorem
Theorem
If f (x) ≤ g (x) ≤ h(x) when x is near c and
limx→c f (x) = limx→c h(x) = L
then limx→c g (x) = L
Evaluate:
1
limx→0 x 2sin( )
x
Ryan Blair (U Penn)
Math 103: One-Sided Limits of Functions
Thursday September 15, 2011
4/7
One-Sided Limits
Definition of One-Sided Limit
Definition
If f (x) is arbitrarily close to L for all x sufficiently close to
c and greater than c, we say f approaches the
rigth-hand limit L as x approaches c and write:
limx→c + f (x) = L
Ryan Blair (U Penn)
Math 103: One-Sided Limits of Functions
Thursday September 15, 2011
5/7
One-Sided Limits
Definition of One-Sided Limit
Definition
If f (x) is arbitrarily close to L for all x sufficiently close to
c and greater than c, we say f approaches the
rigth-hand limit L as x approaches c and write:
limx→c + f (x) = L
Definition
If f (x) is arbitrarily close to L for all x sufficiently close to
c and less than c, we say f approaches the left-hand
limit L as x approaches c and write:
limx→c − f (x) = L
Ryan Blair (U Penn)
Math 103: One-Sided Limits of Functions
Thursday September 15, 2011
5/7
One-Sided Limits
Theorem
limx→c f (x) = L
if and only if
limx→c + f (x) = L and limx→c − f (x) = L.
Ryan Blair (U Penn)
Math 103: One-Sided Limits of Functions
Thursday September 15, 2011
6/7
One-Sided Limits
Theorem
limx→0
Ryan Blair (U Penn)
sin(x)
=1
x
Math 103: One-Sided Limits of Functions
Thursday September 15, 2011
7/7