1.1 A Preview of Calculus 1.1 A Preview of Calculus and and

1.2 Finding Limits 1.2 Finding Limits Graphically and Graphically and

NumericallyNumerically

ObjectivesObjectives

Understand what calculus is and how it Understand what calculus is and how it compares to precalculus.compares to precalculus.

Estimate a limit using a numerical or Estimate a limit using a numerical or graphical approach.graphical approach.

Learn different ways that a limit can fail to Learn different ways that a limit can fail to exist.exist.

Swimming SpeedSwimming Speed

Swimming Speed: Taking it to the LimitSwimming Speed: Taking it to the Limit

Questions 1-5Questions 1-5

Preview of CalculusPreview of Calculus

Diagrams on pages 43 and 44Diagrams on pages 43 and 44

Two Areas of Calculus: Two Areas of Calculus: DifferentiationDifferentiation

Animation of Differentiation Animation of Differentiation

Two Areas of Calculus: IntegrationTwo Areas of Calculus: Integration

Animation of IntegrationAnimation of Integration

LimitsLimits

Both branches of calculus were originally Both branches of calculus were originally explored using limits.explored using limits.

Limits help define calculus.Limits help define calculus.

1.2 Finding Limits Graphically and 1.2 Finding Limits Graphically and NumericallyNumerically

2Graph ( ) 1 with a hole at x=1.

f(x) is not defined at x=1, but it has a limit at 1.

What value does f approach as x gets closer to 1?

f x x x

3 1( )

1

xf x

x

2( 1)( 1)

( )1

x x xf x

x

Find the Limit Find the Limit 3 1

( )1

xf x

x

xx .75.75 .9.9 .99.99 .999.999 11 1.0011.001 1.011.01 1.11.1 1.251.25

f(x)f(x) 2.3132.313 2.7102.710 2.9702.970 2.9972.997 ?? 3.0033.003 3.033.03 3.3103.310 3.8133.813

x approaches 1 from the left

x approaches 1 from the right

1lim ( ) 3xf x

Limits are independent

of single points.

Exploration (p. 48)Exploration (p. 48)

From the graph, it looks like f(2) is defined.From the graph, it looks like f(2) is defined.Look at the table.Look at the table.On the calculator: tblstart 1.8 and On the calculator: tblstart 1.8 and ∆Tbl=0.1.∆Tbl=0.1.Look at the table again.Look at the table again.What does f approach as x gets closer to 2 What does f approach as x gets closer to 2

from both sides?from both sides?

2

2

3 2lim

2x

x x

x

ExampleExample

Look at the graph and the table.Look at the graph and the table.

0lim

1 1x

x

x

0lim 2

1 1x

x

x

ExampleExample

Limits are NOT affected by single points!Limits are NOT affected by single points!

1, 2( )

0, 2

xf x

x

2(2) 0, but lim ( ) 1.

xf f x

Three Examples of Limits that Fail Three Examples of Limits that Fail to Existto Exist

If the left-hand limit doesn't equal right-hand If the left-hand limit doesn't equal right-hand limit, the two-sided limit limit, the two-sided limit does not existdoes not exist..

( )

1, x>0

1, x<0

xf x

x

0 0lim 1 but lim 1 x x

x x

x x

Three Examples of Limits that Fail Three Examples of Limits that Fail to Existto Exist

If the graph approaches If the graph approaches ∞ or -∞ from one ∞ or -∞ from one or both sides, the limit or both sides, the limit does not existdoes not exist..

2

1( )f x

x

From left, as x approaches 0, f(x) approaches .

From right, as x approaches 0, f(x) approaches .

Three Examples of Limits that Fail Three Examples of Limits that Fail to Existto Exist

Look at the graph and table.Look at the graph and table.As x gets close to 0, f(x) doesn't approach As x gets close to 0, f(x) doesn't approach

a number, but oscillates back and forth.a number, but oscillates back and forth.If the graph has an oscillating behaviorIf the graph has an oscillating behavior, ,

the limit the limit does not existdoes not exist..

1( ) sinf x

x

Limits that Fail to ExistLimits that Fail to Exist

f(x) approaches a different number from f(x) approaches a different number from the right side of c than it approaches from the right side of c than it approaches from the left side.the left side.

f(x) increases or decreases without bound f(x) increases or decreases without bound as x approaches c.as x approaches c.

f(x) oscillates as x approaches c.f(x) oscillates as x approaches c.

HomeworkHomework

1.2 (page 54)1.2 (page 54)

#5, 7, #5, 7,

15-23 odd15-23 odd