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Limits I. Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V. Limits Numerically and Graphically VI.Properties of Limits VII.Limits Algebraically VIII.Trigonometric Limits IX.Average and Instantaneous Rates of Change X. Sandwich Theorem XI.Formal Definition of a Limit
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Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Jan 02, 2016

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Hilary Hancock
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Page 1: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

LimitsI. Why limits?II. What are limits?III. Types of LimitsIV. Where Limits Fail to ExistV. Limits Numerically and GraphicallyVI. Properties of LimitsVII. Limits AlgebraicallyVIII. Trigonometric LimitsIX. Average and Instantaneous Rates of ChangeX. Sandwich TheoremXI. Formal Definition of a Limit

Page 2: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Why limits?Limits help us answer the big question of how fast an object is moving at an instant of time. For Newton and Leibniz, this had to do with the velocity a planet moved in its orbit around the sun.

Page 3: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Why limits?We might be more interested in the velocity of other things

Page 4: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Why limits?The fundamental concepts of calculus - the derivative and the integral are both defined in terms of limits. We will see more of these as we learn how to use limits.

b

a

n

kk

n

x

n

abxfdxxf

x

xfxxfxf

1

*

0

)(lim)(

)()(lim)('Derivative

Integral

Page 5: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Why limits?So limits are like the engine under the hood of a car. We are mainly interested in driving the car and won’t spend a lot of time thinking about what is happening under the hood, but we should have a basic understanding of how the engine works.

Page 6: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

What are limits?Limits describe the behavior of functions around specific values of x. They also describe the end behavior of functions.

More specifically, limits describe where the y-value of a function appears to be heading as x gets closer and closer to a particular value or as x approaches positive/negative infinity.

Let’s look at these ideas a little closer.

Page 7: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

What are limits?Some important notes about limits:

a. Limits are real numbers, but we sometimes use to indicate the direction a function

is heading.

x

y

(D.N.E.)Exist Not Does 1

lim0 xx

-1

lim0 xx

11

lim1

xx

Page 8: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

What are limits?Some important notes about limits:

b. Limits do not depend on the value of the function at a specific x value, but on where the function appears to be heading.

x

y

x

y

x

y

1)(lim2

xfx

1)(lim2

xfx

1)(lim2

xfx

Page 9: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

What are limits?c. For a limit to exist, the function must be

heading for the same y-value whether the given x-value is approached from the left or from the right, i.e. one-sided limits must agree.

)(lim)(lim)(lim xfxfxfcxcxcx

x

y

1)(lim2

xfx

2)(lim2

xfx

D.N.E. )(lim)(lim)(lim222

xfxfxfxxx

Page 10: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Types of LimitsThere are three basic forms of limits ( ):

a. Limits at a finite value of x

b. Infinite limits (vertical asymptotes)

c. Limits at Infinity (horizontal asymptotes or end behavior)

Lxfcx

)(lim

)(lim xfcx

Lxfx

)(lim

Lc,

Page 11: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Infinite LimitsInfinite limits occur in the vicinity of vertical asymptotes. Functions may approach positive or negative infinity on either side of a vertical asymptote. Remember to check both sides carefully.

Also, remember to simplify rational expressions before identifying vertical asymptotes.

Page 12: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Infinite Limits Ex. 1Determine the limit of each function as x approaches 1 from the left and from the right.

)(xf )(lim1

xfx

)(lim1

xfx

1

1)(

x

xf

2)1(

1)(

x

xf

1

1)(

x

xf

2)1(

1)(

x

xf

Page 13: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Infinite Limits Ex. 2Identify all vertical asymptotes of the graph of each function.

)1(2

1)(

xxf

1

1)(

2

2

x

xxf

xxf cot)( 4

82)(

2

2

x

xxxf

Page 14: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Limits at Infinity

x

xx

x

x

n

xxxxaxx

lim!limlimlimlnlim

Remember:

Evaluate:!

limx

x x

x

logarithmic polynomial exponential factorial ??????????

Page 15: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Limits at Infinity Video

Page 16: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Limits at Infinity

Page 17: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Limits at Infinity

Page 18: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Where Limits Fail to Exist

There are three places where limits do not exist:

D.N.E. 1

coslim0

xxD.N.E.

5

5lim

0

x

xx

D.N.E. 3

1lim

3 xx

Jump Discontinuities Vertical Asymptotes Oscillating Discontinuities

Page 19: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Limits Numerically 1

Use the TblSet (with Independent set to ASK) and TABLE functions on your graphing calculator to estimate the limit.

Page 20: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Limits Numerically 2

Use the TblSet (with Independent set to ASK) and TABLE functions on your graphing calculator to estimate the limit.

Page 21: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Limits Numerically 3

Use the TblSet (with Independent set to ASK) and TABLE functions on your graphing calculator to estimate the limit.

Page 22: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Limits Graphically 1

Page 23: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Limits Graphically 2

Page 24: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Limits Graphically 3

Page 25: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Limits Graphically 4

Page 26: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Properties of Limits 1

Some examples:?3lim

2

x?lim

4

x

x?lim 2

2

x

x

Thinking graphically may help here.

Page 27: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Properties of Limits 1 Ex. 133lim

2

x

x

y

3)( xf

Page 28: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

xxf )(4lim4

x

x

x

y

Properties of Limits 1 Ex. 2

Page 29: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

2)( xxf 4lim 2

2

x

x

x

y

Properties of Limits 1 Ex. 3

Page 30: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Properties of Limits 2

Page 31: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Properties of Limits 2 Ex. 1 Use the information provided here to evaluate limits a – d here

Page 32: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Properties of Limits 2 Ex. 2 Use the information provided here to evaluate limits a – d here

Page 33: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Properties of Limits 3

For example, evaluate the limit

Page 34: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Properties of Limits 4

For example:

Page 35: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Properties of Limits 5

For example, evaluate the limit

Page 36: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Properties of Limits 6

For example, given:

Evaluate the limit:

Page 37: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Limits AlgebraicallyIn addition to Direct Substitution, there are many strategies for evaluating limits algebraically. In particular, we will focus on three of them:

I.Factor and CancelII.Simplifying FractionsIII.Rationalization

Page 38: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Factor and Cancel

Page 39: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Simplifying Fractions

Basic Strategy:

Multiply numerator and denominator by 3(3+x) and then simplify.

You could also find a common denominator for both fractions in the numerator and then simplify that first.

Page 40: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Rationalization

Page 41: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Trigonometric LimitsThere are two special trigonometric limits:

Page 42: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Trigonometric Limit Ex. 1

Basic strategy:

x

xx

cos

sintan

Page 43: Limits I.Why limits? II.What are limits? III.Types of Limits IV.Where Limits Fail to Exist V.Limits Numerically and Graphically VI.Properties of Limits.

Trigonometric Limit Ex. 2

Strategy: Multiply numerator/denominator by 4

Let and note that as , xy 4 0x 0y