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Math Help - problem with limits

  1. #1
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    problem with limits

    I had some math problems that was troublesome, and I would like to know what are the steps to those problems. Here are the problems: 1)lim x->0 (tan5x/tan2x) the ans. is 5/2. 2) lim x->0 (1-cos(x))/sin(x). The ans. is 0. 3) lim x->0 (sec x-1)/(x sec(x)) the ans. is 0
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  2. #2
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    Quote Originally Posted by driver327 View Post
    I had some math problems that was troublesome, and I would like to know what are the steps to those problems. Here are the problems: 1)lim x->0 (tan5x/tan2x) the ans. is 5/2. 2) lim x->0 (1-cos(x))/sin(x). The ans. is 0. 3) lim x->0 (sec x-1)/(x sec(x)) the ans. is 0
    Problem 1]
    \frac{\tan 5x}{\tan 2x} = \frac{\sin 5x}{\sin 2x} \cdot \frac{\cos 2x}{\cos 5x}

    We can ignore the second factor because \lim_{x\to 0}\frac{\cos 2x}{\cos 5x} = 1 and does not change the value of the limit.

    Thus, the problem reduces to find,
    \lim_{x\to 0}\frac{\sin 5x}{\sin 2x}

    Note, we can write,
    \frac{\sin 5x}{\sin 2x} = \frac{5}{2} \cdot \frac{\sin 5x}{5x} \cdot \frac{2x}{\sin 2x}

    Now, \lim_{x\to 0}\frac{\sin 5x}{5x} = \lim_{x\to 0}\frac{\sin x}{x} =0

    Similar reasoning says that, \lim_{x\to 0}\frac{2x}{\sin 2x} = 1

    Thus, we are left with 5/2.
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  3. #3
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    Quote Originally Posted by driver327 View Post
    2) lim x->0 (1-cos(x))/sin(x). The ans. is 0.
    Hint: Write \frac{1-\cos x}{\sin x} = \frac{1-\cos x}{x} \cdot \frac{x}{\sin x}

    lim x->0 (sec x-1)/(x sec(x))
    Hint: Multiply denominator and numerator by \cos x then apply the argument above.
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    re: lim x->0 (sec x-1)/(x sec(x))

    I did mult. the cos(x) and it got me (1-cos(x)/x) = 0. Did I do this right? One more question, is it a matter of only getting sines and cosines in the equation like this? And thanks PerfectHacker 4 helping me out.
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  5. #5
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    Quote Originally Posted by driver327 View Post
    I did mult. the cos(x) and it got me (1-cos(x)/x) = 0. Did I do this right?
    Yes
    One more question, is it a matter of only getting sines and cosines in the equation like this? And thanks PerfectHacker 4 helping me out.
    Usually.
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