MAT 1234Calculus I

Section 2.3 Part I

Using the Limit Laws

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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.

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

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

Direct Substitution Property

If f(x) 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 f(x) 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 f(x) is a polynomial, then

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

)()(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

Example 1 (Polynomial)

Remark 1

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

Example 2 (Rational Function, a in the domain)

3 is in the domain of the rational function

2

3

6lim

5x

x

x

Example 2 (Rational Function, a in the domain)

2

3

6lim

5x

x

x

3 is in the domain of the rational function

Direct Substitution Property

Can be extended to other functions such as n-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 x=a when it is defined.

)(lim xfax

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

Example 3

3 3 2

1lim 8x

x x

Q&A

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

A: Try the following techniques

Example 4 (Simplify)

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.

Reminders

1

lim 1x

x

1

lim 1

1 1x

x

Reminders

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

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

ba ba

ba ba

2 2

a b

a b

Example 7 (Multiply by conjugate)

0

2 2limh

h

h

2 2

a b a b

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.