Asymptotic

Dec 2008
62
3
If \(\displaystyle f_1(x)\sim f_2(x) \) and \(\displaystyle g_1(x)\sim g_2(x) \), is \(\displaystyle f_1(x)+g_1(x)\sim f_2(x)+g_2(x) \)?
 
Feb 2010
422
141
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
Nov 2009
4,563
1,566
Berkeley, California
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 \(\displaystyle \lim_{x\to\infty}\frac{f(x)}{g(x)}=C\)?
 
Dec 2008
62
3
What does \(\displaystyle \sim\) mean? that \(\displaystyle \lim_{x\to\infty}\frac{f(x)}{g(x)}=C\)?
\(\displaystyle p(x)\sim q(x) \iff \lim_{x\to\infty}\frac{p(x)}{q(x)}=1 \)
 
Sep 2008
481
87
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
\(\displaystyle \dfrac{f_{1}(x)+g_{1}(x)}{f_{2}(x)+g_{2}(x)}=\dfrac{1+g_{1}(x)/f_{1}(x)}{\big(f_{2}(x)/f_{1}(x)\big)+\big(g_{2}(x)/f_{1}(x)\big)}\)
........................\(\displaystyle \sim\dfrac{1+\big(g_{1}(x)/f_{1}(x)\big)}{1+\big(g_{2}(x)/f_{1}(x)\big)}\)
........................\(\displaystyle =\dfrac{\big(f_{1}(x)/g_{1}(x)\big)+1}{\big(f_{1}(x)/g_{1}(x)\big)+\big(g_{2}(x)/g_{1}(x)\big)}\)
........................\(\displaystyle \sim\dfrac{\big(f_{1}(x)/g_{1}(x)\big)+1}{\big(f_{1}(x)/g_{1}(x)\big)+1}\equiv1\).