.

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