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

- anonymous

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

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- chestercat

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- Psymon

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

- anonymous

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

- Psymon

\[\frac{ \frac{ 1 }{ 2}-cosx }{ 2x-\frac{ 2\pi }{ 3 } }\]
Just want to make sure I get it right.

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## More answers

- anonymous

yes

- anonymous

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

- happinessbreaksbones

psymon is so smart D:

- anonymous

@happinessbreaksbones IKR!

- happinessbreaksbones

he's amazing

- happinessbreaksbones

and I'm glad he helped you :)

- anonymous

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

- Psymon

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

- anonymous

the problem I posted?

- Psymon

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.

- anonymous

its ok.

- Psymon

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

- anonymous

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

- terenzreignz

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

- Psymon

That didnt see to do it, though, lol.

- terenzreignz

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

- Psymon

Still gives undefined there xD

- anonymous

that's what I got!

- anonymous

from there I stopped idnt know where to go

- Psymon

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

- anonymous

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

- Psymon

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

- anonymous

yeah true

- hartnn

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

- anonymous

when you do that the answer is 0/0

- hartnn

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

- hartnn

[1/2 - cos (y+pi/3)]/(2y)
easier to deal with ?

- hartnn

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

- terenzreignz

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

- terenzreignz

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

- hartnn

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

- hartnn

i meant 1

- terenzreignz

lol... me too slow ^_^

- hartnn

@have_sabr , are you following us ?
could you expand cos (y+pi/3) ? or did u get till that step?

- anonymous

lol no I have to visualize it.

- hartnn

which step are you now? (stuck at?)

- terenzreignz

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

- terenzreignz

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

- anonymous

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

- terenzreignz

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

- anonymous

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

- hartnn

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

- hartnn

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

- anonymous

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.

- hartnn

just refresh your page...

- Psymon

@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.

- Psymon

|dw:1378282005338:dw|
I just drew what hartnn previously posted. Maybe you can see it when drawn?

- anonymous

oooooh I see.

- Psymon

Now use the sum of cosines formula to expand cos(y+pi/3)

- anonymous

ok

- terenzreignz

Refresher time ^_^
\[\Large \cos(\alpha+ \beta) = \cos(\alpha)\cos(\beta) -\sin(\alpha)\sin(\beta)\]

- anonymous

got it. i expanded it

- Psymon

You can expand it and then separate it into different fractions. Remember, our substitution y is approaching 0 as x approaches pi/3. So once you separate out all your fractions and take y to 0, you'll have your answer.

- anonymous

oooh! I understand thank you all so much!

- Psymon

Lol, thank hartnn xD But glad ya got it :3

- anonymous

THANK YOU HARTNN! I wish there were more ppl like you guys in my class. my class is equally confused as I am

- terenzreignz

One sign that you've truly mastered this bit is if you could teach it to your own peers in turn ^_^

- anonymous

Ikr! I tend to understand even more when I explain to others.

- anonymous

lol I'll come back with more calculus problems again sometime. until then thank you All. good night/good morning/after noo :D

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