limits i.why limits? ii.what are limits? iii.types of limits iv.where limits fail to exist v.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

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.

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

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

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.

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.

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

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

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

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,

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.

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

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

Limits at Infinity

x

xx

x

x

n

xxxxaxx

lim!limlimlimlnlim

Remember:

Evaluate:!

limx

x x

x

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

Limits at Infinity Video

Limits at Infinity

Limits at Infinity

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

Limits Numerically 1

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

Limits Numerically 2

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

Limits Numerically 3

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

Limits Graphically 1

Limits Graphically 2

Limits Graphically 3

Limits Graphically 4

Properties of Limits 1

Some examples:?3lim

2

x?lim

4

x

x?lim 2

2

x

x

Thinking graphically may help here.

Properties of Limits 1 Ex. 133lim

2

x

x

y

3)( xf

xxf )(4lim4

x

x

x

y

Properties of Limits 1 Ex. 2

2)( xxf 4lim 2

2

x

x

x

y

Properties of Limits 1 Ex. 3

Properties of Limits 2

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

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

Properties of Limits 3

For example, evaluate the limit

Properties of Limits 4

For example:

Properties of Limits 5

For example, evaluate the limit

Properties of Limits 6

For example, given:

Evaluate the limit:

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

Factor and Cancel

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.

Rationalization

Trigonometric LimitsThere are two special trigonometric limits:

Trigonometric Limit Ex. 1

Basic strategy:

x

xx

cos

sintan

Trigonometric Limit Ex. 2

Strategy: Multiply numerator/denominator by 4

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