Express the repeating decimal 0.5757... as the ratio of two integers without your calculator.

Warm-Up

Express the repeating decimal 0.5757... as the ratio of two integers without your calculator.

Warm-=Up

x = 0.57

100x = 57.57

99x = 57 x = 57/99

𝑥=0.57  

What is Calculus?

There are only 3 main concepts in calculus.

1) The Limit

2) The Derivative

3) The Integral

What is Calculus?

There are only 3 main concepts in calculus.

1) The Limit

2) The Derivative

3) The Integral

4) You will need a graphing calculator.

1-2:Finding Limits Graphically and NumericallyObjectives:•Understand the concept of a limit

•Calculate limits

Important Ideas•Limits are what make calculus different from algebra and trigonometry•Limits are fundamental to the study of calculus•Limits are related to rate of change•Rate of change is important in engineering & technology

Analysis•Slope is a rate of change•Rate of change is constant at every value on a linear f(x)

m=2

f(x)

x

m=3m=2m=1m=-

1

Analysisf(x)

x

•Rate of change is different at every value on a non-linear f(x)• Rate of change is the slope of the tangent line at a point

Important Ideas•The slope of a secant line is an average rate of change•The slope of a tangent line is an instantaneous rate of change at a point

•We know how to calculate average rate of change

Analysis

•The tangent line problem…

Go to Sketchpad

•We don’t know how to calculate instantaneous rate of change

,therefore,

Warm-Up-You need a graphing calculator. I’m using a TI-84.

Put your signature pages in the box

Important IdeaInstantaneousRate of change is different at every point on f(x)

f(x)

x

Limits are used to calculate the slopes of the tangents

Example1. Graph: 2( )f x x

2. Trace to x=2.

3. Zoom in at least 4 times.4. Describe the graph.

Example

Consider

3 1( ) , 1

1

xf x x

x

What happens at x=1?

x .75 .9 .99 .999

f(x)

Let x get close to 1 from the left:

Try This

Consider

3 1( ) , 1

1

xf x x

x

x 1.25 1.1 1.01

1.001

f(x)

Let x get close to 1 from the right:

Try ThisWhat number does f(x) approach as x approaches 1 from the left and from the right?

3

1

1lim 3

1x

x

x

Try This

Graph and

3

1

1

1

xY

x

2

2 1Y x x on the same axes. What is the

difference between these graphs?

3 1( )

1

xf x

x

Why is there a “hole” in the graph at x=1?

Analysis

ExampleConsider3 1

( )1

xf x

x

for ( ,1) (1, ) and

( ) 4f x

for x=1

3

1

1lim

1x

x

x

=?

Try ThisFind: if

1lim ( )x

f x

2( ) 2, 1f x x x

( ) 1, 1f x x

1lim ( ) 3x

f x

Important Idea

The existence or non-existence of f(x) as x approaches c has no bearing on the existence of the limit of f(x) as x approaches c.

Important Idea

What matters is…what value does f(x) get very, very close to as x gets very,very close to c. This value is the limit.

Try This

Find:

f(0)is undefined; 2 is the limit

2( )

1 1

xf x

x

0lim ( )x

f x

Find:

( ) 1, 0f x x

Try This

( ) , 01 1

xf x x

x

f(0) is defined; 2 is the limit

21

0lim ( )x

f x

Warm-Up

Try ThisFind the limit of f(x) as x approaches 3 where f is defined by:

2 , 3( )

3 , 3

xf x

x

3lim ( ) 2x

f x

ExampleGraph and find the limit (if it exists):

3

3lim

3x x

Important IdeaSome limits do not exist. If f(x) approaches as x approaches c, we say that the limit does not exist at c or, sometimes we say the AP Exam says the limit approaches infinity at c.

ExampleFind the limit if it exists:

0limx

x

x

Important Idea

0lim 1

x

x

x

0lim 1

x

x

x

But…

0limx

x

x

Does not exist

Definition

If a function has a limit, the limit from the right must equal the limit from the left.

Example

1.Graph using a

friendly window:

1sin

x

2. Zoom at x=0

3. Wassup at x=0?

Important Idea

If f(x) bounces from one value to another (oscillates) as x approachs c, the limit of f(x) does not exist at c:

Lesson Close

Name 3 ways a limit may fail to exist.

Assignment

Page 54 Problems 1 - 7 odd, 8 – 24 all In class, we will not cover the formal definition of a limit, sometimes called epsilon-delta definition. I’ll talk about it in NMSI tutoring.