Download Squeeze Theorem & Intermediate Value Theorem

SQUEEZE THEOREM &
INTERMEDIATE VALUE
THEOREM
Section 1-3 continued
Properties Of Limits
Trigonometric Functions
1. lim sin( x)  sin( c)
4.
lim csc(x)  csc(c)
2. lim cos(x)  cos(c)
5.
lim sec(x)  sec(c)
lim tan( x)  tan(c)
x c

6.
lim cot(x)  cot(c)
x c
x c
3.

x c
x c
x c
10) find

11) find

lim cos(x)
x 1
cos(x)
lim
x  2 cot( x)
Squeeze (Sandwich) Theorem
f ,g, and h be functions satisfying
f (x)  g(x)  h(x) for all x near c, except
• Let
possibly at c. If lim g(x)  L
x c

Then
lim f (x)  lim h(x)  L
x c


x c
Squeeze Theorem pg 65
12) Show that
1 
lim x sin   0
x 0
x 
2
1 
1  sin  1
x 

1  2
x  x sin   x
x 
2

2
lim x 2  0
lim x 2  0
x 0

x0
1 
lim x sin   0
x 0
x 
2
13) Find
sin x
lim
x 0
x
graphically
Special Trig Limits
1)
sin( x)
lim
1
x 0
x
Memorize
These!!
2)
1 cos(x)
lim
=0
x 0
x
2
14) Find lim sin (x)
x 0


x
sin( x)  sin( x)
 lim
x 0
x
sin( 3x)
15) find lim
x 0
x

Intermediate Value Theorem
• A function
y  f (x) that is exist for all real
numbers x in the closed interval [a,b] takes on
every value between

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 g ( x )  0 and g ( x)  0
16) Use the IVT to find the value c if the function
f (x)  x  3x  2
3
exists for all real numbers on [0,1] and f (c)  0

17) If g(x) exists for all real x (continuous)
and g(1)  2,g(2)  0,g(3)  4
which of the following g(x) values must
exist on the domain (3,1)


I.
3
2
5
2

II.


III.
3
Homework
Page 67 # 27-34, 65-69 all
Worksheet 1-3