Transcript Document

1.
Warm-Up 1/30
12 ℎ
=
13 8
B
Rigor:
You will learn how to use limits to determine
continuity and describe end behavior of functions.
Relevance:
You will be able to use functions to solve real
world problems.
1-3 Continuity, End
Behavior, and Limits
Continuous Function: has no breaks, holes or gaps.
Limit: is the concept of approaching a value without
necessarily ever reaching it.
Discontinuous Functions: have breaks, holes or gaps.
↑ Nonremovable Discontinuities ↑
Example 1: Determine whether 𝑓 𝑥 = 2𝑥 2 − 3𝑥 − 1 is continuous
at x = 2.
1. Does 𝑓 2 exist?
𝑓 2 =2 2
2
−3 2 −1
𝑓 2 =1
2. Does lim 𝑓 𝑥 exist?
𝑥→2
x
1.9
1.99
1.999
f(x)
0.52
0.95
0.995
2.0
2.001
2.01
2.1
1.005
1.05
1.52
lim 𝑓 𝑥 = 1
𝑥→2
3. Does lim 𝑓 𝑥 = 𝑓(2)?
𝑥→2
1=1
𝑓 𝑥 = 2𝑥 2 − 3𝑥 − 1 is continuous at x = 2.
Example 2a: Determine whether each function is continuous at the
given x-value(s). If discontinuous identify type of discontinuity.
1. Does 𝑓 −3 exist? 𝑓 −3 = 2 − −3
𝑓 −3 = 5
2. Does lim 𝑓 𝑥 exist?
𝑥→−3
lim 𝑓 𝑥
𝑥→−3
does not exist.
𝑓 𝑥 is discontinuous at x = – 3. It has a jump discontinuity.
Example 2b: Determine whether each function is continuous at the
given x-value(s). If discontinuous identify type of discontinuity.
−3 + 3
0
𝑓 −3 =
=
2
−3 − 9 0
1. Do 𝑓 −3 & 𝑓 3 exist?
3 +3
6
𝑓 3 =
=
2
3 −9 0
𝑓 −3 & 𝑓(3) do not exist.
2. Does lim 𝑓 𝑥 exist?
𝑥→−3
lim 𝑓 𝑥 = −.16
𝑥→−3
Does lim 𝑓 𝑥 exist?
𝑥→3
lim 𝑓 𝑥
𝑥→3
does not exist.
𝑓 𝑥 is discontinuous at x = – 3 and x = 3 . It has a removable
discontinuity at x = – 3 and an infinite discontinuity at x = 3 .
Example 3: Determine between which consecutive integers the
real zeros of each function are located on the given interval.
a. 𝑓 𝑥 = 𝑥 3 − 4𝑥 + 2; [– 4, 4]
Zeros between f(– 3) = – 13 and f(– 2) = 2
between f(0) = 2 and f(1) = – 1
between f(1) = – 1 and f(2) = 2
There are real zeros between – 3 and – 2, 0 and 1, and 1 and 2.
b. 𝑓 𝑥 = 𝑥 2 + 𝑥 + 0.16; [– 3, 3]
Zeros between f(– 1) = 0.16 and f(0) = 0.16
There are real zeros between – 1 and 0.
Example 4: Use the graph of the function to describe its end
behavior. Support the conjecture numerically.
𝑓 𝑥 = −𝑥 4 + 8𝑥 3 + 3𝑥 2 + 6𝑥 − 80
lim 𝑓(𝑥) = −∞
lim 𝑓(𝑥) = −∞ and 𝑥→∞
𝑥→−∞
Support Numerically
This supports the conjecture.
Example 5: Use the graph of the function to describe its end
behavior. Support the conjecture numerically.
𝑥
𝑓 𝑥 = 2
𝑥 − 2𝑥 + 8
lim 𝑓(𝑥) = 0
𝑥→−∞
and lim 𝑓(𝑥) = 0
𝑥→∞
Support Numerically
This supports the conjecture.
−1
math!
1-3 Assignment: TX p30-31, 4-40 EOE
Test Corrections Due Wednesday 2/5