I will admit this is a homework problem, but I'm seriously stuck. I'm not looking for answers, but just any hints as to what to do next. Any tips would be appreciated.
I am given:
$$f_1(x) = O(g_1(x)): \exists c_1>0 \; \exists x_1 \; \forall x > x_1 \quad \lvert f_1(x)\rvert \le c_1 \cdot \lvert g_1(x)\rvert$$
$$f_2(x) = O(g_2(x)): \exists c_2>0 \; \exists x_2 \; \forall x > x_2 \quad \lvert f_2(x)\rvert \le c_2 \cdot \lvert g_2(x)\rvert$$
And I have to do the following:
Assume $g_1$ and $g_2$ are non-negative. Find explicit formulas for $c_3$ and $x_3$ (in terms of $c_1, c_2, x_1,$ and $x_2$) so that $$\text{for all } x > x_3, \lvert f_1(x) + f_2(x) \rvert \le c_3 \cdot \lvert g_1(x) + g_2(x) \rvert$$
Essentially I am showing that $f_1(x) + f_2(x) = O(g_1(x) + g_2(x))$.
From the given information I can get that $$ {\lvert f_1(x) \rvert \over \lvert g_1(x)\rvert} \le c_1 \quad \text{and} \quad {\lvert f_2(x) \rvert \over \lvert g_2(x)\rvert} \le c_2$$
But this is where I get stuck. I have no idea how to proceed next. Can anyone give some pointers?