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Thread: Differentiable definition problem

  1. #1
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    Differentiable definition problem

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

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

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