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Math Help - Find a derivative using squeeze theorem

  1. #1
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    Find a derivative using squeeze theorem

    (a) State the limit definition of the derivative of a function f at the point x = a.
    (b) Use (a) to find f'(0).

    f(x)={x^2cos(ln|x|) if x =/= 0
    {0 if x=0


    I'm pretty sure you use the squeeze theorem
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  2. #2
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    Hi
    The cosine of any expression is always between -1 and 1
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  3. #3
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    Quote Originally Posted by ryno16 View Post
    (a) State the limit definition of the derivative of a function f at the point x = a.
    (b) Use (a) to find f'(0).

    f(x)={x^2cos(ln|x|) if x =/= 0
    {0 if x=0


    I'm pretty sure you use the squeeze theorem

    The limit definition of f^{'}(x) is
    f^{'}(x)=\lim_{h \rightarrow 0}\Bigg{(} \frac{(x+h)^{2}\cos(ln|x+h|)-x^2\cos(ln|x|)}{h} \Bigg{)}



    Now for a=0, this means

    f^{'}(0)=\lim_{h \rightarrow 0} \Bigg{(} \frac{h^2\cos(ln|h|)}{h} \Bigg{)} (*)


    But since -1<\cos(x)<1 , \forall x
    \lim_{h \rightarrow 0} -h < (*) < \lim_{h \rightarrow 0} h

    Apply squeeze theorem to conclude that f('0) = 0.
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