1. ## Asymptotic

If $\displaystyle f_1(x)\sim f_2(x)$ and $\displaystyle g_1(x)\sim g_2(x)$, is $\displaystyle f_1(x)+g_1(x)\sim f_2(x)+g_2(x)$?

2. If $\displaystyle f_1\lesssim f_2$ and $\displaystyle g_1 \lesssim g_2$, is $\displaystyle f_1+g_1 \lesssim f_2 + g_2$?

3. Originally Posted by mathman88
If $\displaystyle f_1(x)\sim f_2(x)$ and $\displaystyle g_1(x)\sim g_2(x)$, is $\displaystyle f_1(x)+g_1(x)\sim f_2(x)+g_2(x)$?
What does $\displaystyle \sim$ mean? that $\displaystyle \lim_{x\to\infty}\frac{f(x)}{g(x)}=C$?

4. Originally Posted by Drexel28
What does $\displaystyle \sim$ mean? that $\displaystyle \lim_{x\to\infty}\frac{f(x)}{g(x)}=C$?
$\displaystyle p(x)\sim q(x) \iff \lim_{x\to\infty}\frac{p(x)}{q(x)}=1$

5. Originally Posted by mathman88
If $\displaystyle f_1(x)\sim f_2(x)$ and $\displaystyle g_1(x)\sim g_2(x)$, is $\displaystyle f_1(x)+g_1(x)\sim f_2(x)+g_2(x)$?
It is.
We have to show that $\displaystyle \dfrac{f_{1}(x)+g_{1}(x)}{f_{2}(x)+g_{2}(x)}\sim1$.
and we know that $\displaystyle \dfrac{f_{1}(x)}{f_{2}(x)}\sim1$ and $\displaystyle \dfrac{g_{1}(x)}{g_{2}(x)}\sim1$.
Therefore, we have
$\displaystyle \dfrac{f_{1}(x)+g_{1}(x)}{f_{2}(x)+g_{2}(x)}=\dfra c{1+g_{1}(x)/f_{1}(x)}{\big(f_{2}(x)/f_{1}(x)\big)+\big(g_{2}(x)/f_{1}(x)\big)}$
........................$\displaystyle \sim\dfrac{1+\big(g_{1}(x)/f_{1}(x)\big)}{1+\big(g_{2}(x)/f_{1}(x)\big)}$
........................$\displaystyle =\dfrac{\big(f_{1}(x)/g_{1}(x)\big)+1}{\big(f_{1}(x)/g_{1}(x)\big)+\big(g_{2}(x)/g_{1}(x)\big)}$
........................$\displaystyle \sim\dfrac{\big(f_{1}(x)/g_{1}(x)\big)+1}{\big(f_{1}(x)/g_{1}(x)\big)+1}\equiv1$.