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Math Help - Limits and l'hopitals rule

  1. #1
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    Limits and l'hopitals rule

    I need help with:


    Limit (cos3x)^5/x as x approaches 0

    Should I use L'Hopital's Rule?
    How do i begin to solve this?
    I need help with the first few steps?
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    You should use nothing, there's no indeterminate form.
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  3. #3
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    \lim_{x \to 0} [\cos(3x)]^{\frac{5}{x}}

    let y = [\cos(3x)]^{\frac{5}{x}}

    \ln{y} = \frac{5}{x} \cdot \ln[\cos(3x)]

    \lim_{x \to 0} \frac{5\ln[\cos(3x)]}{x}

    L'Hopital ...

    5 \lim_{x \to 0} {\frac{-3\sin(3x)}{\cos(3x)}} = 5 \cdot 0 = 0

    \lim_{x \to 0} \, \ln{y} = 0

    so ...

    \lim_{x \to 0} \, y = e^0 = 1
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  4. #4
    Math Engineering Student
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    Ahh, that's what happens when they don't use a proper parenthesis.
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