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Math Help - A simple limit problem.

  1. #1
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    A simple limit problem.

    Hello.
    I have to find the limit as x approaches 0 of the function
    [ [(1/(3+x)] -(1/3) ] / x i tried direct substitution and to rationalize the denominator and numerator, but neither works...

    what shoul i do?
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  2. #2
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    Quote Originally Posted by guidol92 View Post
    Hello.
    I have to find the limit as x approaches 0 of the function
    [ [(1/(3+x)] -(1/3) ] / x i tried direct substitution and to rationalize the denominator and numerator, but neither works...

    what shoul i do?
    \frac{\frac{1}{3 + x} - \frac{1}{3}}{x} = \frac{\frac{3 - (3 + x)}{3(3 + x)}}{x}

     = \frac{-\frac{x}{3(3 + x)}}{x}

     = -\frac{1}{3(3 + x)}.


    So \lim_{x \to 0}\left(\frac{\frac{1}{3 + x} - \frac{1}{3}}{x}\right) = \lim_{x \to 0}\left[-\frac{1}{3(3 + x)}\right]

     = -\frac{1}{3(3)}

     = -\frac{1}{9}.
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  3. #3
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    Quote Originally Posted by guidol92 View Post
    Hello.
    I have to find the limit as x approaches 0 of the function
    [ [(1/(3+x)] -(1/3) ] / x i tried direct substitution and to rationalize the denominator and numerator, but neither works...

    what shoul i do?
    It can also be recognised as having the same form as the derivative from first principles of f(t) = 1/t evaluated at t = 3.
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