Question

Let f(x)= sinx/x

a.) What is the average rate of change of f on the interval [π/2, 3π/2]?

b.) What is lim f(x) as x approaches 0?

c.)Is the line x=0 a vertical asymptote of f? Justify your answer using limits.

d.) Use the Squeeze Theorem to show the one y=0 is a horizontal asymptote of f.

Answer 1

average rate of change of f(x) in the interval (a,b) is: f(b)-f(a)/b-a

Answer 2

How do I know what f (π/2) is though?

Answer 3

f(π/2) = sin π/2/π/2 = ?

Answer 4

1/π/2

Answer 5

Sorry, I'm missing my calculator right now -.-

Answer 6

right: 2/π

Answer 7

Ah, yes.

Answer 8

do the same for the other point

Answer 9

-2/3π?

Answer 10

For sin(3π/2)/(3π/2)

Answer 11

OH wait.... 3π/2 isn't even on the unit circle.

Answer 12

it's the angle

Answer 13

yeh it is, never mind...

Answer 14

Yes, -2/3π

Answer 15

f(3π/2) =-2/3π

Answer 16

now do the ratio f(b)-f(a)/b-a

Answer 17

Okay hold on.

Answer 18

-7/2

Answer 19

(-2/3π - 2/π)/(3π/2-π/2)=-8/3π/π=-8/3

Answer 20

sigh... Okay :(

Answer 21

The limit as X approaches 0 would be 1 correct?

Answer 22

yes

Answer 23

X=0 is not a vertical asymptote is it?

Answer 24

right

Answer 25

How can I justify that using limits?

Answer 26

Because x approached infinity from both sides?

Answer 27

infinity and negative infinity that is.

Answer 28

no, lim f(x) for x->0 is 1, remmeber

Answer 29

Yeh but how does that prove that x=0 is not a vertical asymptom?

Answer 30

asymptote*

Answer 31

when a lione is asymptot, the graph should stay in one of its sides, but in this case it is not true

Answer 32

OKay!

Answer 33

Can you give me a brief easy ideal of the "squeeze Theorem". i didn't really understand it in class.

Answer 34

imagine 3 functions: f(x), g(x) and h(x) that have this property:
f(x)< g(x) < h(x)
when x->a limits of f(x) and g(x) are for example = b , so this will be also the limit of g(x)

Answer 35

Hmmm... okay, so how can I use this Theorem to show that y=0 is a horizontal asymptote of f?

Answer 36

http://www.wolframalpha.com/input/?i=f%28x%29%3D+sinx%2Fx
i don't think y=0 is a horizontal asymptot

Answer 37

So it would appear. Very interesting. Perhaps the worksheet has made an error.

Regardless, thanks for your help!

Answer 38

you wellcome