2.1 FINITE LIMITS One Sided Limits, Double Sided Limits and Essential Discontinuities Mathgotserved.com

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2.1 FINITE LIMITS

One Sided Limits, Double Sided

Limits and Essential Discontinuities

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Geometric View of Limits

• Look at a polygon inscribed in a circle

As the number of sides of the polygon increases, the polygon is getting closer to becoming a circle.

If we refer to the polygon as an n-gon, where n is the number of sides we can make

some mathematical statements:

• As n gets larger, the n-gon gets closer to being a circle

• As n approaches infinity, the n-gon approaches the circle

• The limit of the n-gon, as n goes to infinity is the circle

lim( )n

n go circlen

The symbolic statement is: Note: The n-gon approaches a circle in appearance even though it is not a circle. It might as well be a circle.


Definition (verbal)

The limit of a function is the value the function approaches as the independent variable approaches a value, negative or positive infinity. Or think of it as the y value the function approaches as you approach a specific x value.

The Limit of a Function

lim

x a

f x L

Analytical Representation

“Read as the limit of f(x) as x approaches a

is L”Can also be expressed as f(x)→L

as x→a

Properties of Limits

lim lim lim

x c x c x c

f x g x f x g x1. Sum Rule:

lim lim lim

x c x c x c

f x g x f x g x2. Difference Rule:

lim lim limx c x c x c

f x g x f x g x

3.Product Rule:

limlim

lim

x c

x cx c

f xf x

g x g x4. Quotient

Rule:

lim

x c

k k5. Constant Rule:

lim lim

rr ss

x c x cf x f x6. Power Rule


WHY LEARN LIMITS?

Limits are used in derivatives when mathematicians are calculating the velocity of an object in flight.

Real World Application : Calculating Speed(Mathematicians: 42-145k/yr)


One-Sided Limits (Verbal)Numbers x near c fall into two natural categories: those that lie to the left of c and those that lie to the right of c. We write[The left-hand limit of f(x) as x tends to c is L.]

to indicate that as x approaches c from the left, f(x) approaches L.

[The right-hand limit of f(x) as x tends to c is L.]

to indicate that as x approaches c from the right, f(x) approaches L

limx c

f x L

limx c

f x L

We write


How close can Raul Get to the garage?

limraul wall

distance from wall = 0

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Definition of a Limit

Left hand Limit:

We say the limit as x approaches a from the left is

limx a

Lf x

L

x

y

L --

| a

We say the limit as x approaches a from the left is

limx a

Lf x

Right hand Limit:

Double sided Limits

L

limx a

f x L

We say the limit as x approaches a is L

x a x a

Procedures for Finding Finite Limits Algebraically

1. Establish the fact that the limit is finite. (i.e The value that x and the f(x) approaches must not be )

2. Substitute the value x is approaching and evaluate. If 0 is in the denominator go to 3.

3. Factor, or rationalize the numerator or denominator and cancel out any removable discontinuities and substitute again. If 0 at the bottom use a non algebraic approach since the limit might not exist.

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Finding Finite Limits Algebraically

3 2

11. lim 3 5

xx x

Determine the limit of the following

3 23 1 5 1 2

21

12. lim

1x

x

x

1

1 1

x

x x1

x

1x 1x

1

1

x

1

1

1

13. .lim

1

1.lim

1

1.lim

1

x

x

x

xa

x

xb

x

xc

x

1

1

x

x1

1

1

x

x1

1

1

x

x 1 Does not exist


0

sin 3limx

x

x

Problem 4:Determine the limit

Solution:0

sin 3limx

x

x

0

sinlim 1x

x

x

sin3

3

3x

x

0

sin 3lim3

3

x

x

x

0

sin 33lim

3

x

x

x

3 1 3


0

sin 2lim

sin 3x

x

x

Problem 5:Determine the limit

Solution:

0

sinlim 1x

x

x

0

sin 2lim

sin 3x

x

x 0

sin 2lim

sin

2

3

2

33

x

xxxxx

x0

sin 22

2lim

sin 33

3

x

xx

xx

xx

2

x

3 x 0

sin 22

limsin 3

3

x

xx

xx

0

0

sin 2lim

2 2sin 33

lim3

x

x

xx

xx

2 1

3 1 1

Procedure for Finding Graphical Limits

Point: Find the y coordinate of the point that you are approaching from that direction.

Horizontal Asymptote: Find the y coordinate of the equation of the line

Vertical Asymptote: One sided +/- infinity, Double sided does note exist.

0

sinlim

x

x

x

Problem 1:

From both sides you are approaching the point (1,0). The limit is the y coordinate so

-3π -5π/2 -2π -3π/2 -π -π/2 π/2 π 3π/2 2π 5π/2 3π

1

x

y

Solution

0

sinlim 1

x

x

x


Finding Limits Graphically

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

Problem 2:Use the graph to find the following

4a) lim ( )

xg x

4b) lim ( )

xg x

4c) lim ( )

xg x

0d) lim ( )

xg x

0e) lim ( )

xg x

2g) lim ( )

xg x

2h) lim ( )

xg x

2i) lim ( )

xg x

0f) lim ( )

xg x

( )y g x

2

8

DNE

4

4

4

2

2

DNE


f(2) =

lim ( )x

f x

2lim ( )

xf x

4

lim ( )x

f x

2

lim ( )x

f x

2

lim ( )x

f x

4

lim ( )x f x 4

f(4) =

Problem 3

dne

4

4

4

und

1

2

2


Finding Finite Limits Numerically Problem 1:Find the following numerically using the table of values x f (x)

4 5.001

3 5.0001

2 34

1 5.0001

0 5.001

-1 5.01

2a) lim ( )

xf x

2b) lim ( )

xf x

2c) lim ( )

xf x

d) (2)f

5

5

5

34


Problem 2:Find the following limits Numerically

x f (x)

2.02 -9.99

2.01 -9.999

2 Error

1.99 2.9999

1.98 2.999

1.97 2.99

2a) lim ( )

xf x

2b) lim ( )

xf x

2c) lim ( )

xf x

d) 2f

3

10

DNE

Und


Problem 3:Find the limit

2a) lim ( )

xf x

2b) lim ( )

xf x

2c) lim ( )

xf x

d) 2f

2 , if 2

21 if 2

1 1 if 22

x x

f x x

x x


Problem 4:Find the limit

0a) lim ( )

xf x

0b) lim ( )

xf x

0c) lim ( )

xf x

d) 0f

23 1 if 0

4 if 0

x xf x

x x

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2.1 FINITE LIMITS One Sided Limits, Double Sided Limits and Essential Discontinuities Mathgotserved.com