Help with this simple limit question

Question

We have this definition:

$$f(h)=O(h^p) \quad\text{if} \quad \lim_{h\rightarrow0}\frac{f(h)}{h^p}=K\neq 0$$

show this:

$$O(h^2)+O(h^3 )=O(h^2)$$

So what we need to do is follow the definition:

$$\lim_{h\rightarrow0}\frac{f(h)+g(h)}{h^2}=\lim_{h\rightarrow0}\frac{f(h)}{h^2}+\lim_{h\rightarrow0}\frac{g(h)}{h^2}=K_1+\lim_{h\rightarrow0}\frac{g(h)}{h^2}$$

because we know this:

$$\lim_{h\rightarrow0}\frac{f(h)}{h^2}=K_1\neq 0$$ $$\lim_{h\rightarrow0}\frac{g(h)}{h^2}=K_2\neq 0$$

Now we need to give an argument to show $$K_1+\lim_{h\rightarrow0}\frac{g(h)}{h^2}\neq 0$$ with the information: $$\lim_{h\rightarrow0}\frac{g(h)}{h^2}=K_2\neq 0$$

some help please!

Answer 1

Well, $$ \frac{O(h^2)+O(h^3)}{h^2} = \frac{O(h^2)}{h^2}+\frac{O(h^3)}{h^2} \to K + 0 $$ with $K \neq 0$. Indeed, $$ \frac{O(h^3)}{h^2} = \frac{O(h^3)}{h^3} h \to \tilde{K} \cdot 0. $$