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Math Help - Trigonometric limit problem

  1. #1
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    Trigonometric limit problem

    \lim_{x \to a} \frac{sqrt[tanx] - sqrt[tana]}{\sqrt[3]{sinx} - \sqrt[3]{sina}}

    Can anybody help?

    Thank you.
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  2. #2
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    Any idea?
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  3. #3
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    Quote Originally Posted by GreenMile View Post
    \lim_{x \to a} \frac{sqrt[tanx] - sqrt[tana]}{\sqrt[3]{sinx} - \sqrt[3]{sina}}

    Can anybody help?

    Thank you.
    Try multiplying by \frac{x-a}{x-a} to get

    \lim_{x\to a} \frac{\sqrt{\tan{x}}-\sqrt{\tan{a}}}{x-a} \cdot \frac{x-a}{\sqrt[3]{\sin{x}} - \sqrt[3]{\sin{a}}}

    Notice that both can be evaluated using the definition of the derivative at a point:
    f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}
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