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Math Help - Limits and Deivatives

  1. #1
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    Limits and Deivatives

    I need help on the following questions. Please help if you can - I'm stuck.

    Evaluate the limit (if it exists) as x goes to -6
    lim x app -6 6 + 25x + 4x^2/(-36 + 6x + 2x^2)
    The answer has to be in fraction form I'm told

    Differentiate the following function with respect to x
    g(x) = - 3/2 + 6/(5x^12/5) + 2x^9

    Differentiate the following function with respect to x
    f(x) = 9x^4 + (13x^10)/6
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  2. #2
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    #1) So we have the limit: \lim_{x \rightarrow -6} \frac{4x^2+25x+6}{2x^2+6x-36}. Direct substitution gives 0/0. Uh oh. This is a perfect time to use L'Hopital's Rule, which for this case means that since the limit gave an indeterminate form, or 0/0, then the limit is actually equal to the derivative of the numerator divided by the derivative of the denominator. Thus the original limit is equal to \frac{ \frac{d}{dx}(4x^2+25x+6)}{ \frac{d}{dx}(2x^2+6x-36)}. Now try re-evaluating your limit.
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  3. #3
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    Or if you don't know L'Hopital's rule yet, try factoring as \frac{4x^2+25x+6}{2x^2+6x-36} = \frac{(4x+1)(x+6)}{2(x+6)(x-3)} = \frac{4x+1}{2(x-3)} and then use direct substitution.
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