SQUEEZE THEOREM & INTERMEDIATE VALUE THEOREMSection 1-3 continued

Properties Of Limits

1.

2.

3.

4.

5.

6.

Trigonometric Functions

limx c

sin(x) sin(c)

limx c

cos(x) cos(c)

limx c

tan(x) tan(c)

limx c

cot(x) cot(c)

limx c

sec(x) sec(c)

limx c

csc(x) csc(c)

10) find

11) find

limx1

cos(x)

limx

2

cos(x)

cot(x)

Squeeze (Sandwich) Theorem

•Let be functions satisfying

for all x near c, except possibly at c. If

Then

f ,g, and h

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

limx cf (x) lim

x ch(x) L

limx cg(x) L

Squeeze Theorem pg 65

12) Show that

limx 0x 2 sin

1

x

0

1sin1

x

1

x 2 x 2 sin1

x

x 2

limx 0

x 2 0

limx 0x 2 0

limx 0x 2 sin

1

x

0

x

xx

sinlim

013) Find graphically

Special Trig Limits

1)

2)

x

xx

)sin(lim

0

limx 0

1 cos(x)

x

MemorizeThese!!

= 0

1

14) Find

limx 0

sin2(x)

x

limx 0

sin(x) sin(x)

x

limx 0

sin(3x)

x15) find

Intermediate Value Theorem

• A function that is exist for all real numbers x in the closed interval [a,b] takes on every value between

y f (x)

f (a) and f (b) on (a,b)

A continuous function can not skip values

Intermediate Value Theorem

An intuitive example

An airplane takes off and climbs from 0 to 10,000 ft

At some point the planes altitude was exactly 8371 ft.

Existence of a zero: if and 0)( xg 0)( xg

16) Use the IVT to find the value c if the function

exists for all real numbers on

[0,1] and f (c) 0

f (x) x 3 3x 2

17) If exists for all real x (continuous) and

which of the following values must exist on the domain

I. II. III.

g(x)

g( 1) 2,g( 2) 0,g( 3) 4

g(x)

( 3, 1)

3

2

5

2

3

Homework

Page 67 # 27-34, 65-69 all

Worksheet 1-3