Question

fin the lim as x approaches π/3 of 1/2-cosx over 2x-2π/3

help please!

Answer 1

\[\frac{ 1 }{ 2 }-\frac{ cosx }{ 2x }-\frac{ 2\pi }{ 3 }\]?

Answer 2

fin the lim as x approaches π/3 of 1/2-cosx over 2x-2π/3

Answer 3

\[\frac{ \frac{ 1 }{ 2}-cosx }{ 2x-\frac{ 2\pi }{ 3 } }\]

Just want to make sure I get it right.

Answer 4

yes

Answer 5

my teacher told us the answer was square root of 3 over 4. but he never showed the work:/

Answer 6

psymon is so smart D:

Answer 7

@happinessbreaksbones IKR!

Answer 8

he's amazing

Answer 9

and I'm glad he helped you :)

Answer 10

yeah me too. or I woul've been staring at one problem for hours
and still feeling blank.

Answer 11

Im trying to find the calc 1 way of doing this. I can do it the calc 2 way, haha

Answer 12

the problem I posted?

Answer 13

Yeah. I can get the limit through a trick, but its not something you're supposed to know. Im multitasking, so I apologize for the wait.

Answer 14

its ok.

Answer 15

Still trying to take a look. The answer you said before is right, just need to get it a different way, lol.

Answer 16

lol my AP Cal teacher is crazy he gives us the answer and tells us to go home and work out the problem, and that we must know for the quizzes he's given out the next day

Answer 17

Those fraction bars look really intimidating... can't there be just one? :3

Answer 18

That didnt see to do it, though, lol.

Answer 19

\[\Large \frac{ \frac{ 1 }{ 2}-cosx }{ 2x-\frac{ 2\pi }{ 3 } }= \frac{\frac{1-2\cos(x)}{2}}{\frac{6x-2\pi}{3}}\]\[\Large = \frac{3-6\cos(x)}{12x -4\pi}\]

Answer 20

Still gives undefined there xD

Answer 21

that's what I got!

Answer 22

from there I stopped idnt know where to go

Answer 23

That x - pi is whats miserably getting in the way.

Answer 24

for the numerator I did simplify and got 3(1-2cos)

Answer 25

Thats not the problem, its the x - pi. Thats what our focus needs to be on.

Answer 26

yeah true

Answer 27

i would say , just substitute y = x- pi/3 and simplify first...

Answer 28

when you do that the answer is 0/0

Answer 29

the answer will always be 0/0, unless we cancel out a common factor which makes num and denom=0

Answer 30

[1/2 - cos (y+pi/3)]/(2y)

easier to deal with ?

Answer 31

expand cos (y+pi/3) now, can you ?

Answer 32

What I did... combines elements of Master Hartnn's idea of substituting \[\Large y = x-\frac{\pi}3\]

Answer 33

And the fact that
\[\Large \lim_{p\rightarrow 0}\frac{1-\cos(p)}{p}=0\]

Answer 34

and the fact that lim sin x/x = 0 when x->0

Answer 35

i meant 1

Answer 36

lol... me too slow ^_^

Answer 37

@have_sabr , are you following us ?

could you expand cos (y+pi/3) ? or did u get till that step?

Answer 38

lol no I have to visualize it.

Answer 39

which step are you now? (stuck at?)

Answer 40

Why don't you start with this:
\[\Large = \frac{3-6\cos(x)}{12x -4\pi}\]
And set \[\large x = y+\frac{\pi}3\]

Answer 41

And do the necessary substituting?
By the way,
\[\Large x \rightarrow \frac \pi 3\]\[\implies\]\[\color{blue}{\Large y \rightarrow0}\]

Answer 42

i did what @terenzreignz said set x equal to pi/3

Answer 43

I never said that ^_^
I said set
\[\LARGE x = \color{red}{y+}\frac \pi 3\]

Answer 44

lol omg im to slow to do this over the enternet. i would actually have to see exactly.

Answer 45

\(\large \frac{ \frac{ 1 }{ 2}-cosx }{ 2x-\frac{ 2\pi }{ 3 } }=\frac{ \frac{ 1 }{ 2}-cosx }{ 2(x-\frac{ \pi }{ 3 } )}=\dfrac{\dfrac{1}{2}-\cos(y+\pi/3)}{2y}\)
when x = y+pi/3

Answer 46

got that? now can you expand cos(y+pi/3) ?

Answer 47

the equation was a picture before and its just whole bunch of brackets and number which i didn't get. so you guys thanks for the trouble you went though but i just cant get. I'll see if my teacher ould really explain.

Answer 48

just refresh your page...

Answer 49

@have_sabr Well, we have a bit of it laid out for you. You can expand like hartnn was saying by using the sum of cosines formula.

Answer 50

I just drew what hartnn previously posted. Maybe you can see it when drawn?