Function o(h)

Definition

Function $o(h)$ is defined as a function $f(x)$ such that

$$\underset{h\to 0}{\lim }\frac{f(h)}{h}=0$$

Example

(a) The function $f(x)=x^2$ is $o(h)$ since ${\lim }_{h\to 0}\frac{f(h)}{h}={\lim }_{h\to 0}\frac{h^2}{h}={\lim }_{h\to 0}h=0$

(b) The function $f(x)=x$ is not $o(h)$ since ${\lim }_{h\to 0}\frac{f(h)}{h}={\lim }_{h\to 0}\frac{h}{h}={\lim }_{h\to 0}1=1\ne 0$

(c) If $f(\cdot )$ is $o(h)$ and $g(\cdot )$ is $o(h)$, then so is $f(\cdot )+g(\cdot )$. This follows since ${\lim }_{h\to 0}\frac{f(h)+g(h)}{h}={\lim }_{h\to 0}\frac{f(h)}{h}+{\lim }_{h\to 0}\frac{g(h)}{h}=0+0=0$

(d) If $f(\cdot )$ is $o(h)$, then so is $g(\cdot )=cf(\cdot )$. This follows since ${\lim }_{h\to 0}\frac{cf(h)}{h}=c{\lim }_{h\to 0}\frac{f(h)}{h}=c\cdot 0=0$

(e) From (c) and (d) it follows that any finite linear combination of functions, each of which is $o(h)$, is $o(h)$.