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Math Help - Differentiable definition problem

  1. #1
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    Differentiable definition problem

    Suppose that the function f: \mathbb {R} \rightarrow \mathbb {R} is differentiable at x_0 = 0. Prove that  \lim _{x \rightarrow 0} \frac {f(x^2)-f(0)}{x} = 0

    How should I approach the f(x^2) here?
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by tttcomrader View Post
    Suppose that the function f: \mathbb {R} \rightarrow \mathbb {R} is differentiable at x_0 = 0. Prove that  \lim _{x \rightarrow 0} \frac {f(x^2)-f(0)}{x} = 0

    How should I approach the f(x^2) here?
    \begin{aligned}\lim_{x\to{0}}\frac{f(x^2)-f(0)}{x}&=\lim_{x\to{0}}\frac{f(x^2)-f(0)}{x-0}\\<br />
&=\left[f(x^2)\right]'|_{x=0}\\<br />
&=2xf'\left(x^2\right)|_{x=0}\\<br />
&=0<br />
\end{aligned}
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