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Q:

Evaluating $\frac{f'(x)}{\sqrt{f(x)}}$ at $x=0$

I came across a problem:

Let $f\colon\mathbb{R}\to\mathbb{R}$ be differentiable on $\mathbb{R}$. Then $$\frac{f'(0)}{\sqrt{f(0)}}=\lim_{x\to0}\frac{f(x)-f(0)}{x-0}=\lim_{x\to0}\frac{f(x)}{x}$$

Now, if $f(x)$ tends to $f(0)$ as $x\to0$, then the limit is $1$. But the problem asks for the case when $f(x)\to0$ as $x\to0$. In this case, my intuition tells me that the limit can’t be $1$, because if it were, the above limit would be equal to $1$. I’ve tried proving that the limit can’t be $1$, but I couldn’t reach a conclusion.

A:

We can now see where your mistake is.
Indeed, let $x \in \mathbb{R} \setminus \{ 0 \}$. The mean value theorem then implies that $f(x) – f(0) = f'(c)(x-0)$ for some $c \in (0,x)$. Replacing $x$ with $0$ yields
$$0 = \lim_{x\to0} \frac{f(x)}{x} \quad \iff \quad \lim_{x\to0} f'(c) = 0.$$
However,
$$\lim_{x\to0} f'(c)
eq 0,$$
whereas the limit on the right-hand side is $1$.

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