Asymptotic

mathman88

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

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

Drexel28

MHF Hall of Honor
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 $$\displaystyle \lim_{x\to\infty}\frac{f(x)}{g(x)}=C$$?

mathman88

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

bkarpuz

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
$$\displaystyle \dfrac{f_{1}(x)+g_{1}(x)}{f_{2}(x)+g_{2}(x)}=\dfrac{1+g_{1}(x)/f_{1}(x)}{\big(f_{2}(x)/f_{1}(x)\big)+\big(g_{2}(x)/f_{1}(x)\big)}$$
........................$$\displaystyle \sim\dfrac{1+\big(g_{1}(x)/f_{1}(x)\big)}{1+\big(g_{2}(x)/f_{1}(x)\big)}$$
........................$$\displaystyle =\dfrac{\big(f_{1}(x)/g_{1}(x)\big)+1}{\big(f_{1}(x)/g_{1}(x)\big)+\big(g_{2}(x)/g_{1}(x)\big)}$$
........................$$\displaystyle \sim\dfrac{\big(f_{1}(x)/g_{1}(x)\big)+1}{\big(f_{1}(x)/g_{1}(x)\big)+1}\equiv1$$.