Is the limit of $f/f'$ necessarily $0$?

Let $f:[a,b]\to\mathbb{R}$ be differentiable on $[a,b]$, with $\lim\limits_{x\to a}f(x)=\lim\limits_{x\to a}f^\prime(x)=0$, and $f^\prime(x)\ne 0$ in a neighborhood of $a$. Is it necessarily true that

$$\lim_{x\to a}\frac{f(x)}{f^\prime(x)}=0$$

It doesn't seem necessarily clear to me that this limit should always exist, and that there's not some pathological function for which the limit is nonzero.

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