I am attempting the following question:
assume $$ f(h) = p(h) + O(h^n)$$
and
$$ g(h) = q(h) + O(h^m) $$
as $$h \to 0$$
I am required to find the largest interger r such that
$$f(h) + g(h) = p(h) + q(h) + O(h^r)$$
I am bit confused but what i have done is consider the following :-
$$f(h) + g(h) = p(h) + O(h^n) +q(h) + O (h^m)$$
case where m > n then the following holds
$$f(h) + g(h) = p(h) + q(h) + O(h^m)$$ where m = r
case where m < n then the following holds
$$f(h) + g(h) = p(h) + q(h) + O(h^n)$$ where n = r
I am not sure if this is correct and how to go about evaluating this but i am hoping someone can guide me or correct me.
When $h$ is going toward zero, small powers go to zero more slowly than large powers so they will dominate. As an example, take $f(x)=x+x^2, g(x)=x+x^3, p(x)=q(x)=x$ Then $$f(x)=p(x)+O(x^2)\\ g(x)=q(x)+O(x^3)\\ f(x)+g(x)=2x+x^2+x^3=p(x)+q(x)+x^2+x^3=p(x)+q(x)+O(x^2)$$ because as $x \to 0, x^2 + x^3 \in O(x^2)$ because the $x^3$ becomes negligible. $$r=\min(m,n)$$