Results 1 to 3 of 3

Math Help - limits with trig function test prep

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    76

    limits with trig function test prep

    I am preparing for a test and the following question is from the practice exam. My problem is that the explanation skips over things that I don't understand. I was hoping somebody here could clear it up for me.

    Find the limit:
    lim x(sin 5x) / (1 − cos x) = ?
    x→0

    I know that : x sin x / x = 1
    and : 1-cos x / x^2 = 1/2

    And these come into play in this problem.. .it's the "getting there" I don't understand. here's the solution given:

    lim x(sin 5x) / (1 − cos x) = lim 5x^2(sin 5x) / 5x(1-cosx) = 5/(1/2)
    x→0 x→0

    so the answer is 10.

    The solution never shows where the known things above actually appear. where is the x that x sin x is divided by? Where is the x^2 that 1-cosx is divided by?

    I am sure this is supposed to be understood, but I don't understand it. Please help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,860
    Thanks
    742
    Hello, littlejodo!

    Find: . \lim_{x\to0}\frac{x\sin5x}{1-\cos x}

    I know that: . \lim_{x\to0}\frac{\sin x}{x}\:=\:1\;\text{ and }\;\lim_{x\to0}\frac{1-\cos x}{x^2} \:=\:\tfrac{1}{2} . . . . Good!

    We have: . \frac{x\sin5x}{1-\cos x}

    Divide top and bottom by x^2\!:\quad\frac{\dfrac{x\sin5x}{x^2} }{\dfrac{1-\cos x}{x^2}} \;=\;\frac{\dfrac{\sin 5x}{x}}{\dfrac{1-\cos x}{x^2}}

    Multiply numerator by \frac{5}{5}\!:\quad \frac{\frac{5}{5}\cdot\dfrac{\sin5x}{x}}{\dfrac{1-\cos x}{x^2}} \;=\;\frac{5\cdot\dfrac{\sin5x}{5x}}{\dfrac{1-\cos x}{x^2}}


    Therefore: . \lim_{x\to0}\left[\frac{5\cdot\dfrac{\sin5x}{5x}} {\dfrac{1-\cos x}{x^2}}\right] \;=\; \frac{5\cdot1}{\frac{1}{2}} \;=\;10<br />



    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2008
    Posts
    76
    thank you so much!!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Function, using two-path test, finding limits
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 29th 2010, 07:03 PM
  2. Taylor Series/Limits/Convergence Test
    Posted in the Calculus Forum
    Replies: 4
    Last Post: May 28th 2009, 01:33 PM
  3. new to calculus--test monday--limits
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 31st 2009, 11:45 AM
  4. Prep Qs for test
    Posted in the Algebra Forum
    Replies: 1
    Last Post: December 12th 2008, 03:25 AM
  5. 1st Calculus Test Prep
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 12th 2007, 08:43 PM

Search Tags


/mathhelpforum @mathhelpforum