anonymous
  • anonymous
fin the lim as x approaches π/3 of 1/2-cosx over 2x-2π/3 help please!
Calculus1
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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Psymon
  • Psymon
\[\frac{ 1 }{ 2 }-\frac{ cosx }{ 2x }-\frac{ 2\pi }{ 3 }\]?
anonymous
  • anonymous
fin the lim as x approaches π/3 of 1/2-cosx over 2x-2π/3
Psymon
  • Psymon
\[\frac{ \frac{ 1 }{ 2}-cosx }{ 2x-\frac{ 2\pi }{ 3 } }\] Just want to make sure I get it right.

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anonymous
  • anonymous
yes
anonymous
  • anonymous
my teacher told us the answer was square root of 3 over 4. but he never showed the work:/
happinessbreaksbones
  • happinessbreaksbones
psymon is so smart D:
anonymous
  • anonymous
@happinessbreaksbones IKR!
happinessbreaksbones
  • happinessbreaksbones
he's amazing
happinessbreaksbones
  • happinessbreaksbones
and I'm glad he helped you :)
anonymous
  • anonymous
yeah me too. or I woul've been staring at one problem for hours and still feeling blank.
Psymon
  • Psymon
Im trying to find the calc 1 way of doing this. I can do it the calc 2 way, haha
anonymous
  • anonymous
the problem I posted?
Psymon
  • 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
  • anonymous
its ok.
Psymon
  • Psymon
Still trying to take a look. The answer you said before is right, just need to get it a different way, lol.
anonymous
  • 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
  • terenzreignz
Those fraction bars look really intimidating... can't there be just one? :3
Psymon
  • Psymon
That didnt see to do it, though, lol.
terenzreignz
  • 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
  • Psymon
Still gives undefined there xD
anonymous
  • anonymous
that's what I got!
anonymous
  • anonymous
from there I stopped idnt know where to go
Psymon
  • Psymon
That x - pi is whats miserably getting in the way.
anonymous
  • anonymous
for the numerator I did simplify and got 3(1-2cos)
Psymon
  • Psymon
Thats not the problem, its the x - pi. Thats what our focus needs to be on.
anonymous
  • anonymous
yeah true
hartnn
  • hartnn
i would say , just substitute y = x- pi/3 and simplify first...
anonymous
  • anonymous
when you do that the answer is 0/0
hartnn
  • hartnn
the answer will always be 0/0, unless we cancel out a common factor which makes num and denom=0
hartnn
  • hartnn
[1/2 - cos (y+pi/3)]/(2y) easier to deal with ?
hartnn
  • hartnn
expand cos (y+pi/3) now, can you ?
terenzreignz
  • terenzreignz
What I did... combines elements of Master Hartnn's idea of substituting \[\Large y = x-\frac{\pi}3\]
terenzreignz
  • terenzreignz
And the fact that \[\Large \lim_{p\rightarrow 0}\frac{1-\cos(p)}{p}=0\]
hartnn
  • hartnn
and the fact that lim sin x/x = 0 when x->0
hartnn
  • hartnn
i meant 1
terenzreignz
  • terenzreignz
lol... me too slow ^_^
hartnn
  • hartnn
@have_sabr , are you following us ? could you expand cos (y+pi/3) ? or did u get till that step?
anonymous
  • anonymous
lol no I have to visualize it.
hartnn
  • hartnn
which step are you now? (stuck at?)
terenzreignz
  • 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
  • terenzreignz
And do the necessary substituting? By the way, \[\Large x \rightarrow \frac \pi 3\]\[\implies\]\[\color{blue}{\Large y \rightarrow0}\]
anonymous
  • anonymous
i did what @terenzreignz said set x equal to pi/3
terenzreignz
  • terenzreignz
I never said that ^_^ I said set \[\LARGE x = \color{red}{y+}\frac \pi 3\]
anonymous
  • anonymous
lol omg im to slow to do this over the enternet. i would actually have to see exactly.
hartnn
  • 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
  • hartnn
got that? now can you expand cos(y+pi/3) ?
anonymous
  • 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
  • hartnn
just refresh your page...
Psymon
  • 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
  • Psymon
|dw:1378282005338:dw| I just drew what hartnn previously posted. Maybe you can see it when drawn?
anonymous
  • anonymous
oooooh I see.
Psymon
  • Psymon
Now use the sum of cosines formula to expand cos(y+pi/3)
anonymous
  • anonymous
ok
terenzreignz
  • terenzreignz
Refresher time ^_^ \[\Large \cos(\alpha+ \beta) = \cos(\alpha)\cos(\beta) -\sin(\alpha)\sin(\beta)\]
anonymous
  • anonymous
got it. i expanded it
Psymon
  • 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
  • anonymous
oooh! I understand thank you all so much!
Psymon
  • Psymon
Lol, thank hartnn xD But glad ya got it :3
anonymous
  • 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
  • terenzreignz
One sign that you've truly mastered this bit is if you could teach it to your own peers in turn ^_^
anonymous
  • anonymous
Ikr! I tend to understand even more when I explain to others.
anonymous
  • 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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