MAT 1234Calculus I

Section 1.6 Part I

Using the Limit Laws

http://myhome.spu.edu/lauw

HW

Homework 1.6 Part I Do your HW ASAP. Tutoring is available!!! Seriously, Do not wait.

Quiz Friday and …

Quiz :1.6 Part I (2 problems from your HW)

Make sure you have an approved calculator.

Recall

Limit of the following form is important

1.4: Estimate limits by tables 1.6: Compute limits by algebra 1.5: Formally define limits

h

afhafh

)()(lim

0

Preview

Limit LawsDirect Substitution PropertyPractical summary of all the limit laws

Limit Laws

11 limit laws that help us to compute limits (printed on p.5).

Foundation of computing limits, but tedious to use.

Practical methods will be introduced.

Limit Laws

7. limx ac c

x

y

a

c y c

Limit Laws

8. limx ax a

x

y

a

y x

Limit Laws

If and exist, then )(lim xfax

)(lim xgax

1. lim ( ) ( ) lim ( ) lim ( )

3. lim ( ) lim ( )x a x a x a

x a x a

f x g x f x g x

cf x c f x

Example 1

2

lim 2 5x

x

1. lim ( ) ( ) lim ( ) lim ( )

3. lim ( ) lim ( )

7. lim

8. lim

x a x a x a

x a x a

x a

x a

f x g x f x g x

cf x c f x

c c

x a

2

lim 2 5x

x

Direct Substitution Property

If is a polynomial, then

Also true if f(x) is a rational function and a is in the domain of f

)()(lim afxfax

Direct Substitution Property

If is a polynomial, then

Also true if is a rational function and is in the domain of

,

)()(lim afxfax

Direct Substitution Property

If is a polynomial, then

Also true if is a rational function and is in the domain of

)()(lim afxfax

Why?

Polynomials are “continuous” functions

x

y

a

lim ( ) ( )x a

f x f a

Why?

Polynomials are “continuous” functionslim ( ) lim ( ) ( )x a x a

f x f x f a

x

y

a

( )f a

Why?

Polynomials are “continuous” functions

lim ( ) lim ( )

m

)

l

(

i

x a x af x f x

f xx a

f a

x

y

a

( )f a

Example 1 (Polynomial)

2

lim 2 5x

x

2

lim 2 5x

x

Remark 1

Once you substitute in the number, you do not need the limit sign anymore.

Example 2 (Rational Function, in the domain)

2

3

6lim

5x

x

x

Example 2 (Rational Function, in the domain)

3 is in the domain of the rational function

2

3

6lim

5x

x

x

2

3

6lim

5x

x

x

Direct Substitution Property

Can be extended to other functions such as -th root.

Not for all functions such as absolute value, piecewise defined functions.

Limit Laws Summary

Use Direct Substitutions if possible*. That is, plug in when it is defined.

)(lim xfax

* Sums, differences, products, quotients, -th root functions of polynomials,

Example 3

3 3 2

1lim 8x

x x

3 3 2

1lim 8x

x x

Q&A

Q: What to do if the answer is undefined when plugging in ?

A: Try the following techniques

Example 4 (Simplify)2

1

1lim

1x

x

x

2

1

1lim

1x

x

x

1.Use equal signs

2.Use parentheses for expressions with sums and differences of more than 1 term.

3. Show the substitution step.

Expectations: Standard Notations and Presentation

1

lim 1x

x

1

lim 1

1 1x

x

Expectations: Standard Notations and Presentation

4. Do not actually “cross out” terms.

1

1limx

x

1

1

x

x

Remark 1 Again

Once you substitute in the number, you do not need the limit sign anymore.

1

lim 1

1 1x

x

Example 5 (Combine the terms)

21

1 2lim

1 1x x x

21

1 2lim

1 1x x x

Remark 1 Again (What? Again!)

Once you substitute in the number, you do not need the limit sign anymore.

1

1lim

11

1 1

x x

Example 7 (Multiply by conjugate)

Review of conjugates

The conjugate of is

The conjugate of is

The product of conjugates is

Example 7 (Multiply by conjugate)

0

2 2limh

h

h

2 2

a b a b

a b a b

0

2 2limh

h

h

Review: We learned…

Limit Laws Direct Substitution Property of

polynomials and rational functions Techniques

• Simplify• Combine the terms• Multiply by conjugate

Classwork

Use pencils Use “=“ signs Do not “cross out” anything. Do not skip steps

Once you substitute in the number, you do not need the limit sign anymore.

I Do not Know my Algebra....

What to do?

What would Jesus do?