. Blast from the Past Find point(s) of intersection 1. 2. 3.4. 5.
Transcript of . Blast from the Past Find point(s) of intersection 1. 2. 3.4. 5.
.
Blast from the Past
Find point(s) of intersection
1. 2.
3. 4.
5.
2484
1224
yx
yx192
11
yx
yx
18710
95
yx
yx
2
2
4 xxy
xy
32
342
2
xxy
xxy
.
Blast from the Past
Find point(s) of intersection
1. 2.
3. 4.
5.
2484
1224
yx
yx
192
11
yx
yx
18710
95
yx
yx32
342
2
xxy
xxy
)6,6( )1,10(
)4,1(
)4,2(0,0
)3,0(0,3
2
2
4 xxy
xy
.
Blast from the PastIn exercises 1 – 4 write the inequality in the form
1. 2.
3. 4.
bxa
4x 52 x
71 x 13 x
In exercises 5 and 6 write the fraction in reduced form
5. 6. 3
1832
x
xx
12
22
2
xx
xx
.
Blast from the PastIn exercises 1 – 4 write the inequality in the form
1. 2.
3. 4.
bxa
4x 52 x
71 x 13 x
In exercises 5 and 6 write the fraction in reduced form
5. 6. 3
1832
x
xx
12
22
2
xx
xx
44 x
68 x
73 x
42 xx
6x 1x
x
Finding Limits Graphically and Numerically
Limits are restrictions…mathematically f(x) is restricted by the graph of the function.
In this section you will estimate a limit using a numerical or graphical approach.
Learn different ways that a limit can fail to exist.
Lets analyze the graph of f (x):
.
To get an idea of the behavior of the graph of f (x) near x = 0 lets use a set of values approaching 0 from the left and set of values approaching 0 from the right.
Estimating a Limit Numerically
Slide 2- 7
Analyzing the table we see although x cannot equal 0,we can come
arbitrarily close to 0 and as a result the function f(x) moves arbitrarily close
to 2.
x -0.01 -0.001 -0.0001 -0.00001 0 0.00001 0.0001 0.001 0.01
f(x) 1.9949 1.99950 1.99995 1.99995 2.00005 2.0005 2.00499 2.0049
Which leads to the limit notation:
Which reads as “the limit of f(x) as x approaches 0 is 2
2)(lim0
xfx
.
Exploration:
Estimate numerically the
x 0.75 0.9 0.99 0.999 1 1.001 1.01 1.1 1.25
f(x) 2.313 2.710 2.970 2.997 ? 3.003 3.030 3.310 3.813
X 1.75 1.9 1.99 1.999 2 2.001 2.01 2.1 2.25
f(x) ? ? ? ? ? ? ? ? ?
1
1lim
3
1
x
x
x
Complete the chart and estimate numerically the2
23lim
2
2
x
xx
x
Finding a Limit Graphically
Use the graph of f(x) to determine each limit:
)(lim0
xfx
)(lim1
xfx
)(lim1
xfx
)1(f
Behavior That Differs from Right to Left:
The following graphs of f(x) represent functions that the limit of f(x) as x, approaches a, does not exist.
As x approaches a from the right:
Limits That Fail to Exist
)(lim xfax
As x approaches a from the left:
)(lim xfax
Because f(x) approaches a different number from the left side of a than it approaches from the right side, the limit does not exist.
Unbounded Behavior:
The following graph of f(x) represent function that the does not exist
)(lim0
xfx
xxf
1)(
Oscillating Behavior:
The following graph of f(x) represent function that the does not exist
xxf
1sin)(
)(lim0
xfx
Exit Ticket
1. Write a brief description of the meaning of the notation
2. If , can you conclude anything about the limit of f(x) as x approaches 2? Explain your reasoning.
25)(lim8
xfx
4)2( f
.
Blast from the Past
In exercises 1 and 2 rationalize the numerator
1. 2. x
x 11 4
35
x
x
In exercise 3 simplify expression
3.
4. If then
xx 3
13
1
23)( xxf xxf
.
Blast from the Past
In exercises 1 and 2 rationalize the numerator
1. 2. x
x 11 4
35
x
x
In exercise 3 simplify expression
3.
4. If then
xx 3
13
1
23)( xxf
x
xfxxf
)(
11
1
x 35
1
x
33
1
x
3
5. Use the graph to find the limit (if it exists). If the limit does not exist, explain why.
2lim ( )x
f x
2 2, 2( )
2, 6
x if xf x
if x
.
6. Use the graph to find the limit (if it exists). If the limit does not exist, explain why.
3
3lim
3x
x
x
(0)f
0lim( )x
x
( 2)f
2lim ( )
xf x
7. Use the graph of the function f to decide whether the value of the given quantity exists. If it does, find it. If not explain why.
a.
b.
c.
d.
8. Use the graph of the function f to decide whether the value of the given quantity exists. If it does, find it. If not explain why.
6lim ( )
xf x
( 6)f
( 3)f
3lim ( )
xf x
0lim ( )x
f x
(0)f
a.
b.
c.
d.
e.
f.
Evaluating Limits Analytically
In this section you will evaluate a limit using properties of limits.
Develop a strategy for finding limits.Evaluate a limit using dividing out and rationalizing
techniques.
.
Previous section we learned that does not depend
on the value of the function at x = c.
However, it may happen a limit may be precisely f(c), the limit can be evaluated by Direct Substitution.
These well-behaved functions are continuous.
)(lim xfcx
Analytical Approach
3)( xf 1)( xxf 22)( xxf
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
Determine the limits of the functions
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
)(lim0
xfx
)(lim1
xfx
)(lim1
xfx
Remember a limit may be precisely f(c) and the limit can be evaluated by Direct Substitution if the function is continuous at x = c.
Properties of Limits: Where f(x) and g(x) are functions with limits that exists such that and
1. Scalar multiple: where b is a constant
2. Sum or difference:
3. Product:
4. Quotient:
5. Power:
Indeterminate Form of a Limit
Direct Substitution may produce the solution 0/0, an indeterminate form
Dividing Technique
Rationalizing Technique
Simplifying Technique
3
62
3lim x
xx
x
52lim3
xx
x
x
x
11lim
0 2
1
11
1lim
0
xx
x
x
x sin
tan3lim
0
3cos
3lim
0
xx
Exploration
In exercises 1 – 5 evaluate the limits
1.
2.
3.
4.
5.
453 2
2lim
xxx
2
452
3
3lim
x
xx
x
3
1
5lim xx
1
12
1lim
x
x
x
x
x
x
22lim
0
Limits of Trigonometric Functions
Let a be a real number in the domain of the given trigonometric function.
Two Special Trigonometric Limits
1. 2. 1sin
lim0
x
x
x
0cos1
lim0
x
x
x
Exploration of Limits of FunctionsThe principle of direct substitution can be expanded to include exponential, logarithmic, piece-wise, and inverse trigonometric functions
1. 5.
2. 6.
3. 7.
4. 8.
1,1
1,4)()(
21
limxx
xxxfxf
x
x
x
x
sin3lim
0
x
x
1
21
sinlim
xx
2lnlim1
x
x
x
sinlim
2
x
x
x log
lnlim
2
x
x
x
)cos1(5lim
0
x
x
elim2
.
Exit Ticket
2,5
2,3)(
x
xxf1. Let find
2. Let find
3. What is meant by indeterminate form?
)(lim2
xfx
0,
0,)(
2 xx
xxxf
)(lim
0xf
x
MotionThe height of a ball thrown straight up with an initial speed of 48 ft/sec from a roof top 160 feet high is s(t) = -16 +48t + 160 where t is the elapsed time the ball is in the air.
a) What does the function s(t) represent?
b) What is the initial height of the ball?
c) When does the ball strike the ground?
d) What is the average velocity of the ball between t = 0 and t = 2?
MotionThe height of a ball thrown straight up with an initial speed of 48 ft/sec from a roof top 160 feet high is s(t) = -16 +48t + 160 where t is the elapsed time the ball is in the air.
a) What does the function s(t) represent? The function s(t) represents the position function of the ball.
b) What is the initial height of the ball?The initial height of the ball is 160 feet.
c) When does the ball strike the ground?The ball strikes the ground when s(t)=0, t = 5 sec.
d) What is the average velocity of the ball between t = 0 and t = 2?The average velocity of the ball between t = 0 and t = 2 is 16 ft/sec