Assume I have a function $ f:\mathbb{R}^{m}\to\mathbb{R}^n $, and let $ h\in \mathbb{R}^m $.
Does the notation $ f\left(x_0+h\right)=o\left(h\right) $ (for fixed $x_0 $) mean that$$ \lim_{h\to0}\frac{||f\left(x_{0}+h\right)||_{\mathbb{R}^{n}}}{||h||_{\mathbb{R}^{m}}}=0 ?$$
Because I've seen in a few places that lecturers wrote just $ \lim_{h\to0}\frac{f\left(x_{0}+h\right)}{||h||}=0 $, which doesn't make any sense to me. I'll be glad for a clarification. What's the acceptable definition worldwide? Thanks in advance
$\endgroup$2 Answers
$\begingroup$I would never say $f(x_0 + h) = o(h)$ when $f$ has values in $\mathbb R^n$ rather than $\mathbb R$.
But if I were going to say it, it would mean one of the two limits you wrote. Those limits are equivalent: $\frac{f(x_0 + h)}{\|h\|} \to 0$ (that is, $0 \in \mathbb R^n$) exactly when $\frac{\|f(x_0 + h)\|}{\|h\|} \to 0$ (that is, $0 \in \mathbb R$).
That's because the definition of a limit in $\mathbb R^n$ means that the first of these limits holds if and only if $\left\|\frac{f(x_0 + h)}{\|h\|} - 0\right\| \to 0$, and this simplifies to the second limit.
$\endgroup$ 6 $\begingroup$Writing
$$\lim_{h\to0}\frac{f\left(x_{0}+h\right)}{||h||_{\mathbb R^m}}=0$$ does make sense. For $h \neq 0$, the map $g(h) = \frac{f(x_0+h)}{\Vert h \Vert_{\mathbb R^m}}$ is well defined. Stating that its limit is equal to $0$ means that
$$\lim\limits_{h \to 0} \left\Vert \frac{f(x_0+h)}{\Vert h \Vert_{\mathbb R^m}} - 0\right\Vert_{\mathbb R^n} = \lim\limits_{h \to 0} \left\Vert \frac{f(x_0+h)}{\Vert h \Vert_{\mathbb R^m}} \right\Vert_{\mathbb R^n} = \lim\limits_{h \to 0} \frac{\Vert f(x_0+h) \Vert_{\mathbb R^n}}{\Vert h \Vert_{\mathbb R^M}}=0.$$
So the two definitions are indeed equivalent.
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