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Thread: limits: the limit, as x -> 0, of (1 - cosx) / sinx

  1. #1

    limits: the limit, as x -> 0, of (1 - cosx) / sinx

    I have trouble solving trig limits...

    lim
    x--> 0
    1-cosx
    sinx

    I don't understand why they break up the problem this way... to this:

    (x) (1-cosx)
    (sinx) (x)

  2. #2
    Elite Member
    Join Date
    Sep 2005
    Posts
    7,293

    Re: limits

    Here's one way to do it. Remember this method. It will come in handy with limits. That way you can transform it into something you can work with.

    Multiply top and bottom by 1+cos(x)

    [tex]\lim_{x\to 0}\frac{1-cos(x)}{sin(x)}[/tex]

    [tex]\frac{(1-cos(x))(1+cos(x))}{sin(x)(1+cos(x))}[/tex]

    [tex]=\frac{1-cos^{2}(x)}{sin(x)(1+cos(x))}[/tex]

    The top is equal to [tex]sin^{2}(x)[/tex], so we have:

    [tex]\frac{sin^{2}(x)}{sin(x)(1+cos(x))}[/tex]

    [tex]=\lim_{x\to 0}\frac{sin(x)}{1+cos(x)}[/tex]

    Now, see the limit?. It is easy now.

  3. #3

    Re: limits

    okay. I see, thanks for the tip .

  4. #4
    Junior Member
    Join Date
    Dec 2007
    Posts
    124

    Re: limits: the limit, as x -> 0, of (1 - cosx) / sinx

    Quote Originally Posted by chao2006
    I have trouble solving trig limits...

    lim
    x--> 0
    1-cosx
    sinx

    I don't understand why they break up the problem this way... to this:

    (x) (1-cosx)
    (sinx) (x)
    ------------------------------------------------
    Sorry about the mangling of your question -- it's a total mess and the SYSTEM did that to us.
    Anyway, I think 'they' wanted to test your knowledge of these 'standard' limits:

    sin x
    ----- --> 1
    x
    1 - cos x
    ------------ --> 0 << typo fixed.
    x
    Of course the example can be done without putting in the x's, as the last post showed.

  5. #5
    Elite Member
    Join Date
    Sep 2005
    Posts
    7,293

    Re: limits: the limit, as x -> 0, of (1 - cosx) / sinx

    That's why we have LaTex. Or you can wrap it in 'code' tags to keep it in line.


    Code:
                      sinx
         lim        ------  = 1
       x->0            x

    This looks better though:

    [tex]\lim_{x\to 0}\frac{sin(x)}{x}=1[/tex]

  6. #6
    Elite Member
    Join Date
    Jan 2005
    Location
    Lexington, MA
    Posts
    5,626

    Re: limits: the limit, as x -> 0, of (1 - cosx) / sinx

    Hello, chao2006!

    [tex]\lim_{x\to0} \frac{1-\cos x}{\sin x}[/tex]

    [tex]\text{I don&#39;t understand why they break up the problem this way: }\:\frac{x}{\sin x}\cdot\frac{1-\cos x}{x}[/tex]

    PAULK saw the reason . . .

    We're expected to know these two theorems:

    . . [tex]\lim_{x\to0}\frac{\sin x}{x} \:=\:1 \qquad\qquad \lim_{x\to0}\frac{1-\cos x}{x} \:=\:0[/tex]


    [tex]\text{Then: }\;\lim_{x\to0}\frac{1-\cos x}{\sin x} \;=\;\lim_{x\to0}\left[\frac{x}{\sin x}\cdot\frac{1-\cos x}{x}\right] \;=\;1\cdot 0 \;=\;0[/tex]

    I'm the other of the two guys who "do" homework.

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