1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
August 27, 2013
Warm Up
What is
(A) 0
(B) nonexistent (C) 1 (D) -1
(E) none of these
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
August 27, 2013
1.4 Continuity and One Sided Limits
Objective: You will be able to:
• define continuity
• evaluate one sided limits
• understand and use the Intermediate Value Theorem
(special case of when lim L ≠ lim R)
*finger test*
we want more information of
where left and where right
the limit still DNE)
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
August 27, 2013
Stand and Deliver
A function is continuous at c if...
1. f (c) is defined (the point exists)
2. lim f (x) exists (the limit exists)
x c
3. f (c) = lim f (x) (the point = the limit)
x c
1.4
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
August 27, 2013
Three conditions exist for which the graph of f is not continuous at x = c.
1.
Why is this not continuous?
not continuous because f(c) is not defined
(the point DNE)
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
2.
August 27, 2013
Why is this not continuous?
not continuous because lim f(x) DNE
x c
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
3.
August 27, 2013
Why is this not continuous?
not continuous because f(c) ≠ lim f(x)
x c
point DNE the limit 1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
http://www.calculus­help.com/continuity/
(ok)
August 27, 2013
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
Discontinuities can be ...
1. Removable
2. Nonremovable
August 27, 2013
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
August 27, 2013
1. Removable: only discontinuities at one spot and
could be "fixed" by redefining f (c)
ex.
x2 ‐ 4
f(x) =
x‐2
f(x)= x+2
twin function
looks the same in all
but one point
{
if f(x) =
x2 ‐ 4
x‐2
4
,x≠2
,x=2
then it would be
continuous
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
2. Nonremovable
Big gaps or asymptotes, cannot be "fixed"
ex.
f(x) =
1
x
August 27, 2013
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
August 27, 2013
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
August 27, 2013
One Sided Limits
For cases where x approaches different
values from the right and left, we can look at
one sided limits.
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
August 27, 2013
One Sided Limits
A limit from the left is denoted lim f(x) = L
x c­
A limit from the right is
denoted lim f(x) = L
x c+
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
Calculus in Motion:
Limits.gsp
(fair)
August 27, 2013
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
f(x) =
[x [
lim f(x) =
x 2-
lim f(x) =
x 2+
August 27, 2013
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
August 27, 2013
Classifying Discontinuities
Removable or point
(Holes)
2 sided limit exists
Essential or Non­removable
Jump
1 sided limits exist
Infinite (vertical asymptotes)
at least one of the 1 sided limits don't exist
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
August 27, 2013
The Intermediate Value Theorem (page 77)
If f is continuous on the closed interval [a, b]
and k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = k.
f is continuous on [a, b]. There exist a c such that f(c) = k.
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
http://www.calculus­help.com/tutorials
intermediate value theorem
(good)
August 27, 2013
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
August 27, 2013
Use the Intermediate Value Theorem to explain why the function has a zero in the given interval:
f(x) = x3 + 3x ­ 2 [0,1]
f(0) = f(1) = 1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
August 27, 2013
EXIT
TICKET
A function is continuous if...
1.
2.
3.
and / or...
The graph of the function f is shown.
Which of the following statements
about f is true?
(A)
(B)
(C)
(D)
(E) DNE
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
AP Question
August 27, 2013
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
August 27, 2013
1.4 Continuity, one sided limits, Intermediate Value Theorem.notebook
August 27, 2013
Attachments
Limits.gsp