Unit 1 Limits. Slide 2 1.1 Limits Limit – Assume that a function f(x) is defined for all x near c...

Unit 1

Limits

1.1 LimitsLimit – Assume that a function f(x) is defined for all x near c (in some open interval containing c) but not necessarily at c. We say that, “the limit of f(x) as x approaches c is equal to L”

* c and L are

limx cf x L

1.1 LimitsExistence of a Limit

– The functions value must the same value

from

– If a function is not defined on one side of c, the limit

1.1 ExamplesFind f(-2),

1) 2)

22 2

lim , lim , lim .xx x

f x f x f x

4

2

-2

-4

-5

4

2

-2

-4

-5

1.1 ExamplesFind f(-2),

3) 4)

22 2

lim , lim , lim .xx x

f x f x f x

4

2

-2

-4

-5

4

2

-2

-4

-5

1.1 Examples

4

2

-2

-4

-5 5

4

3

3

1

1

3

5

5

5) lim

6) lim

7) lim

8) lim

9) lim

10) lim

11) lim

12) lim

x

x

x

x

x

x

x

x

f x

f x

f x

f x

f x

f x

f x

f x

The graph of f(x) is shown below. Use the graph to evaluate each limit.

1.1 Examples

Piecewise Function – A function obtained by together

multiple functions on .

Sketch the graph and then evaluate each limit.

2

2

3, for 1 3 , for 1 2

5 , for 2

x xf x x x

x x x

1

1

1

2

13) lim

14) lim

15) lim

16) lim

x

x

x

x

f x

f x

f x

f x

6

4

2

-2

-4

-6

-5 5

1.1 LimitsEvaluating limits without graphing

1) Substitute the value you are into x.

*If you get a definite answer, then the limit equals

.

*If you get , we have more work to do.

2) the expression.

*

*Multiply by

3)

00

1.1 ExamplesDetermine the value of each limit without graphing.

2

22

2 317) lim

5 14x

x xx x

518) lim

xe


22

2 419) lim

12 3t

tt

3

2

420) lim

2x

x xx


2

4

2 921) lim

4h

h hh

3

1

122) lim

1x

xx


16

423) lim

16x

xx

0

4 224) lim

h

hh


23

31

1) lim9x

xx

2

22) lim

4x

xx x


3 2

2

143) lim

2x

x x xx

1.3 More Evaluating LimitsEvaluating limits

If you substitute and get , then the limit equals

either or .

non-zero #0

Evaluate each limit.

5

5

5

41)

5lim

lim

lim

x

x

x

f xx

f x

f x

f x

1.3 ExamplesEvaluate each limit.

2

2

2

2

32)

2

lim

lim

lim

x

x

x

xf x

x

f x

f x

f x

2

2

1

1

1

23)

1lim

lim

lim

x

x

x

xf x

xf x

f x

f x

1.3 ExamplesInvestigate left and right hand limits, then evaluate.

3

21

14) lim

1x

xx

2

22

25) lim

6y

y yy y

1.3 Examples6) Sketch a possible graph of f(x).

2

2

4

0 1 2 3 4 3

lim 1

lim 3

lim 2

x

x

x

ff ff f

f x

f x

f x

6

4

2

-2

-4

-6

-5 5

1.4 Limits to Infinity

Limit of f as x approaches infinity – means the limit of f as x moves

increasingly far to the .

Limit of f as x approaches negative infinity – means the limit of f as

x moves increasingly far to the .

1limx x

1lim

x x

2

1limx x

2

1lim

x x

1.4 ExamplesEvaluate each limit without graphing.

2

5 2

20 31) lim

3 4 5x

x xx x

27 92) lim

4 3x

xx

1.4 ExamplesEvaluate each limit without graphing.

2

23) lim1x

xx

4) lim cosx

x

1.5 Vertical and Horizontal AsymptotesVertical Asymptote

If , then f(x) has a vertical

asymptote at .

If , then f(x) has a vertical

asymptote at .Horizontal Asymptote

If , then f(x) has a horizontal asymptote

at .

If , then f(x) has a horizontal asymptote

at .

limx a

f x

limx a

f x

limx

f x

limx

f x

1.5 ExamplesGive the equations of all asymptotes.

31)

5x

f xx

2

22)

4f x

x

1.5 ExamplesGive the equations of all asymptotes.

3 2

2

2 33)

2 6 8x x x

f xx x

2

2

2 154)

9x x

f xx

1.5 Examples5) Sketch a graph of f(x).

1 1

lim 3, lim 3

lim , limx x

x x

f x f x

f x f x

1.6 Continuity

A function f is continuous at x = c if and only if the following three

conditions are met.

1)

2)

3)

1

-1

0.5

1.6 ContinuityFour types of discontinuities.1) 2)

3) 4)

4

2

-2

-4

-5

4

2

-2

-4

-5

4

2

-2

-4

-5

1.6 ExamplesFind each point of discontinuity and identify the type.

2

11)

4 3x

f xx x

2

22)

3 2x

f xx x

1.6 ExamplesFind each point of discontinuity and identify the type.

2 5, for 3

3) 2, for 3 1, for 3

x xf x x

x x

2 3, for 2

4)7 , for 2x x

f xx x

1.6 Examples5) Explain why f is not continuous at x = 2.

2 3 10

, for 22

4, for 2

x xx

f x xx

1.6 Examples6) Find the value of k that will make f continuous at x = –3.

2 5 24

, for 33

k, for 3

x xx

f x xx

1.6 Examples7) Find the value of c that will make f continuous at x = –1.

2 5, for 1

3, for 1cx x

f xcx x

1.6 ContinuityIntermediate Value Theorem

If f is continuous on [a, b] and w is between f(a) and f(b),

then there exists at least one c on [a, b] such that .

1.7 Rate of Change

Average Rate of Change

Amount of change divided by the time it takes.

*

1) Find the average rate of change of on [5, 21]. 4f x x

1.7 Rate of ChangeSketch the graph of f(x) = 2x² on [0, 5].

Calculate the average rate ofchange for each interval.2) [2, 4]

3) [2, 3]

4) [2, 2.5]

5) [2, 2.01]

50

40

30

20

10

2 4

1.7 Rate of Change

What were we really finding?

*

*To calculate: find the limit of the slope of the secant line as the

points get closer and closer together.Write this limit using the points (a, f(a)) and (a + h, f(a + h))

1.7 Rate of Change6) Find the instantaneous rate of change of f at x = 5.

3 1f x x

1.7 Rate of Change7) Find the slope of the tangent line at x = a.

5f x x

1.7 Rate of ChangeNormal Line

*8) Write equations of the tangent and normal lines at x = -1.

2 5 6f x x x

1.8 Definition of the DerivativeWe use the derivative to help us find the slope of the tangent line.

Notation:

Prime:

: dydx

1) If f(6) = -5 and f’(6) = 2, write equations of the tangent and normal lines at x = 6.

1.8 Definition of the Derivative

Definition of the Derivative:The derivative of a function f is another function f’ whosevalue at x is

provided this limit exists.Alternate Form:

0

limh

f a h f ah

' lim

x a

f x f af a

x a

1.8 Examples f(x) = x² + 3x - 1

2) Use the definition of the derivative to find f’(-3).

3) Use the alternate form of the derivation to find f’(2).

1.8 Examples y = 3x² - 10x + 2

4) Find y’ at x = a.

5) Find y’ at x = 3.

1.8 Examples

6) Write equations of the tangent and normal lines at x = 3.

7) The graph of f(x) is shown below. Sketch the graph of f’(x).

Unit 1 Limits. Slide 2 1.1 Limits Limit – Assume that a function f(x) is defined for all x near c...

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Transcript of Unit 1 Limits. Slide 2 1.1 Limits Limit – Assume that a function f(x) is defined for all x near c...