MAT 1234Calculus I

Section 1.6 Part II

Using the Limit Laws

http://myhome.spu.edu/lauw

Homework

WebAssign 1.6 Part II Turn in the one problem at the end of

your handout tomorrow. Friday special again this Friday ->OMH 215

Recall: Limit of a Function

if and only if

and

Independent of f(a)

Lxfax

)(lim

Lxfax

)(lim Lxfax

)(lim

Preview

Limits with Absolute Values The Squeeze Theorem Pay extra attention to the presentation

expectations.

Recall (absolute value function)

0 if

0 if

xx

xxx

Example 8(a) (Limits with Absolute Values)

xx 0lim

This is not a polynomial or an rational function

Absolute function is not in one of the 11 limit laws

Find the limit by using the definition of limits

Example 8(a) (Limits with Absolute Values)

xx 0lim

This is not a polynomial or an rational function

Absolute function is not in one of the 11 limit laws

Find the limit by using the definition of limits

Example 8(a) (Limits with Absolute Values)

xx 0lim

This is not a polynomial or an rational function

Absolute function is not in one of the 11 limit laws

Find the limit by using the definition of limits

Example 8(a) (Limits with Absolute Values)

xx 0lim

This is not a polynomial or an rational function

Absolute function is not in one of the 11 limit laws

Find the limit by using the definition of limits

Example 8(a) (Limits with Absolute Values)

We are going to look at left and right hand limits:

1.

2.

xx 0lim

As 0 , 0x x

As 0 , 0x x

if 0

if 0

x xx

x x

0limx

x

0limx

x

0limx

x

Example 8(a) (Limits with Absolute Values) Note that solutions steps are very important

here. You need to pay attention and understand the correct way of presenting your solutions.

WebAssign do not check your solutions steps.

Advise: Write out full solutions in your notebook even WebAssign does not ask for it.

Example 8(b)

2

2lim

2x

x

x

Example 8(b)

2

2lim

2x

x

x

Again, no direct substitution!!

if 0

if 0

2 if 2 02

( 2) if 2 0

x xx

x x

x xx

x x

1.

2.

Example 8(b)

2

2lim

2x

x

x

2 if 22

( 2) if 2

x xx

x x

As 2 , 2x x

As 2 , 2x x

2

2lim

2x

x

x

2

2lim

2x

x

x

Example 9 (Limits with Absolute Values)

0

1 1lim 5x x x

Example 9 (Limits with Absolute Values)

0

1 1lim 5x x x

if 0

if 0

x xx

x x

As 0 , 0x x

0

1 1lim 5x x x

Squeeze Theorem

Lxg

Lxhxf

axxhxgxf

ax

axax

)(lim then

)(lim)(lim and

near is when )()()( If

Squeeze Theorem

( ) ( ) ( )f x g x h x

x

y

a

)(xf

)(xg

)(xh

Squeeze Theorem

x

y

L

a

)(xf

)(xh( ) ( ) ( )f x g x h x

lim ( ) lim ( )x a x a

f x h x L

Squeeze Theorem

x

y

L

a

)(xf

)(xg

)(xh( ) ( ) ( )f x g x h x

lim ( ) lim ( )x a x a

f x h x L

lim ( )x ag x L

Squeeze Theorem

You will see

this type of

idea over and

over again.

x

y

L

a

)(xf

)(xg

)(xh

Example 10

We cannot apply the limit laws since

DNE (Lab 2, 4(c))

xx

xx

xxx

1sinlimlim

1sinlim

0

2

0

2

0

xx

1sinlim

0

Example 10

We cannot apply the limit laws since

DNE (Lab 2, 4(c))

xx

xx

xxx

1sinlimlim

1sinlim

0

2

0

2

0

xx

1sinlim

0

Example 10

We cannot apply the limit laws since

DNE (Lab 2)

xx

xx

xxx

1sinlimlim

1sinlim

0

2

0

2

0

xx

1sinlim

0

Example 10

1siny

x

Example 10

( ) ( ) ( )f x g x h x

lim ( ) lim ( )x a x a

f x h x L

lim ( )x ag x L

1sin

x

2

0

1lim sinxx

x

2 1sinx

x

Expectations

1sin

x

2 1sinx

x

0

0

lim

limx

x

and

Example 10

2 1siny x

x

Review: We learned…

How to find limits involving absolute values. The idea of the squeeze theorem. Pay attention to the expectations of the solutions

details. Note that in the exam, you may be ask to

state the theorems you learned in this class.

Quiz

Please use pencils and erasers. If your handwriting is difficult to

understand,… You have 20 min. Do not look at your neighbor’s paper. If

you need to relax your neck, … Double check your work. Once you turn in your paper, you can go.

Quiz

0

2 2Evaluate lim

h

h

h

0

2 2limh

h

h