3-2 Limits

Limit as x approaches a number

x #lim

Just what is a limit?

A limit is what the ___________________________

__________________________________________

__________________________________________

__________________________________________

We may not actually get there. BUT a limit is not what you actually get to, but appear to get to.

Asymptote: _________________________

Lets look at a couple Graphs

3y x

1y

x

2

1y

x

What is the way to solve it?

The easiest way to find the limit value is to plug the number in.

Find the following

x 31. lim 4x

x

2x 52. lim

3

x 2

13. lim

x

What about

What is the restriction? (what can’t the denominator be)

Factor the top and see if any terms cancel out

If a term cancels out _______________________________________________________________

Graph the above equation

2x 5x 6x 2

What if….

What do you think the answer is if you plug

in the number and get ?

What do you think the answer is if you plug

in the number and get ?

0#

#0

The Harmonic SequenceThe process used to find limits as x is based

on

the Harmonic Sequence

The is 0. Think about it. What about

As x gets really really huge, what will the reciprocal of the fxns approach?

1

f(x)x

x

1lim

x

x

3lim

x x 2

3lim

x

x 5

9lim

x

Some rules of limits

The great thing about limits is that the limit of something complicated can be done as the limit of all the pieces.

xlim (A B) _____________

xlim (A B) _____________

xlim (A B) _____________

xlim (A B) _____________

Taking the limit of Equations

Steps to Solve:

Divide each term by the highest overall power you see in the problem. Then evaluate each of the pieces. Then take the limit of each term.

2lim 3 4x x x

2

x 2

x 2x 3 lim

x x

Example

Group Problems

Find the following

x 2

2x 53. lim

x 3

3

x 2

x 5x 24. lim

x 3x 1

x

5x1. lim

x 4

2

x 2

x 2x 32. lim

x x

Now there is a shortcut “trick” to these problems. WITHOUT TALKING TO ANYONE tonight see if you can figure it out.

3-2 Limits

Day 2

Anyone figure out the short cut?That’s right. You look for the overall high power.

Overall High Power in top = _______________

Overall High Power in bottom = ____________

High Power in top and bottom are the same

=_________________________________

__________________________________

Examples of Graphs

So, you can see that the graphs have these vertical and horizontal lines that act as boundary lines. These are called Asymptotes. __________________________________________________________________________________________________________________

AsymptotesVertical Asymptotes (VA): ___________________________________________________________

Hole: ___________________________________

Why does it still count if it goes away? ________________________________________________________________________________

Horizontal Asymptotes (HA): ________________________________________________________

Step 1 – Find Vertical and Horizontal Asymptotes

1

1. yx 2

x

2. yx 1

2x 43. y

x 2

2

x 24. y

x 3x 4

Step 2 - Plot 3 points on each side of the vertical asymptote(s).

• Graph

1

1. yx 2

x

2. yx 1

2x 43. y

x 2

2

x 24. y

x 3x 4

2

x 24. y

x 3x 4

-1 4

**Graphs can cross a Horizontal Asym but not a vert Asym.

3-3 Oblique Asymptotes

OK – to review for just a minute

If a VA cancels out, ___________________

___________________________________

If a VA doesn’t cancel out, ______________

___________________________________

FYI: _______________________________

___________________________________

So what is an oblique?

Did you notice that all of the graphs that had vertical asymptotes also had limits? That is, the only functions that didn’t have limits had holes.

What if you have no limit to the function, but as well have a vertical asymptote?

Such as, Lets go graphing! 2x 2x 3x

2x 2x 3x

What happened?

Well the vertical asymptote stayed, but the graph didn’t level. There was a diagonal line that acted as a boundary line. This diagonal line is the __________________

___________________________________

So, lets figure out how to find the OA.

Finding OB Asymptote

When the limit does not exist and there is a restriction, _________________________

___________________________________

To find the OA, ___________________ into the numerator and ignore the remainder. That ______ is the oblique asymptote. Then graph the function the same way as if there was a VA.

Cancels

Take the Limit

Check denominator of

F(x)

Plot points and graph the function

There will be a hole

Exists Plot 3 points on each side of VA

Divide Denominator into

Numerator

DNE

Doesn’t

Plot the OA as a dashed line; then plot 3 points on

either side” of the OA

2x 2x 31.

x

Graph the following

2

2x 2x 12.

x

Graph the following

2

2x x 13.

x 1

Group Problem

1

1

3.4 Day 1

Solving Fractional Equations

1. Find the x intercept of the graph

X-int? ________________________

y 3x 21

Practice solving Equations

x 2 1

x 6 x x 6

Isolating Variables:

2ab6. Isolate b: c=

a+b

ab

7. Isolate b r=a+b+c

1 20

3

x

x

Solving the Inequalities

•______________________________

_______________________________

•______________________________

•______________________________

Example

50

2

x

x

3-4 Day 2

Word Problems

Work Rate Problems

•____________________________________

_____________________________________

_____________________________________

_____________________________________

•___________________________________

•____________________________________

Jan can tile a floor in 14 hours. Together, Jan and her helper can tile the same floor together in 9 hours. How long would it take Bill to do the job alone?

Work Rate x Time = Work done

1.The denominator of a fraction is 1 less than twice the numerator. If 7 is added to both numerator and denominator, the resulting fraction has a value of 7/10. Find the original fraction.

Examples

Example

A student received grades of 72, 75 and 78 on three tests. What must he score on the next test to average a 80?

3.6 Synthetic Division

What is Synthetic Division?

Synthetic Division ____________________

___________________________________

1. ________________________________

2. ________________________________

3. ________________________________

___________________________________

___________________________________

Lets try a problem

Please divide by long division.

3 2x x 11x 12x 2

This is Synthetic Division

This is the equivalent problem in synthetic division form:

___________________________________

___________________________________

2 1 1 -11 +12

This is synthetic Division

Try synthetic Division and see what you get:

1 4 3

1 4 3 -3

1 1 3 -1 -6

Here’s another problem with a bit of a twist.

If your last name begins with A-M, do this problem by long division.

If your last name begins with N-Z, do this problem by synthetic division.

3 22x x 3x 72x 1

What did you notice?

Answer is doubled.

When there is a number in front of the x, ________________________________________________________________________________________________________________________________________

**One other rule

If one of the x’s are missing plug a zero in its place!

3.6 Day 2Why Synthetic Division?

What use is this method, besides the obvious saving of time and

paper?

The Remainder Theorem

If is not a factor of F(x), then

___________________________________

___________________________________

That is

x c

long synth R R F(c)

The Factor Theorem

If is a factor of F(x) then _________

When we talk about roots, it’s the same as zeros. Set equal to zero and solve.

x c

How does this apply?

1. Find F(2) if

2. Is (x – 2) a factor of

2F(x) x 17x 41

5 2F(x) x 16x 3

4 3 2F(x) x 3x 2x 12x 8; 2

Factor Completely and find the roots:

Find the polynomial that has as roots 1, -1 and 7

1

1

7

x

x

x

Polynomial = ________________

3.7 Solving Polynomial Equations

That is, finding all the roots of P(x)

without a head start

There are 5 main rules we will use to determine possible rational roots. There are others that you can read about in the book, but these 5 are the basic ones you narrow down the possibilities.

Remember: when you divide synthetically, if the remainder = 0 _______________________. If the remainder ≠0 then the number is not a root and never will be.

Rule #1

The only possible real rational roots are

Where

pq

p ___________________

q ________________________

Rule #2

If the signs of all the terms in the polynomial are +, ____________________________

___________________________________

Think about this, using

2P(x) 2x 7x 3

Rule #3

If the signs of the terms of the polynomial alternate 1 to 1 (that is + – + – + –) then __________________________________

___________________________________

If a term is missing, it is ok to assign it a + or – value to make it fit this rule.

Rule #4

If you add all the coefficients and get 0, then 1 is a root. Otherwise 1 is not a root (and never will be a root ever).

This is a good one. Essentially if it works, you have your start point.

Rule #5

Change the signs of the odd powered coefficients and then add. If you get 0, then -1 is a root. Otherwise, -1 is not a root.

Sometimes this one isn’t worth the effort.

Why these rules?

You will have a list of possibilities (and maybe a definite) with which to start synthetically dividing.

Remember – the goal of the problem is to find all zeroes (or factor). A zero is something whose factor divides evenly into a function. Therefore, synthetically you want to get a remainder of 0.

3 21. F(x) x 3x 3x 1

4 3 22. F(x) 2x 5x x 4x 4