Asymptotic

**mathman88** · June 8th 2010, 03:38 PM

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

**maddas** · June 8th 2010, 03:45 PM

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

**Drexel28** · June 8th 2010, 03:45 PM

Originally Posted by mathman88

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

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

**mathman88** · June 8th 2010, 04:44 PM

Originally Posted by Drexel28

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

$p(x)\sim q(x) \iff \lim_{x\to\infty}\frac{p(x)}{q(x)}=1$

**bkarpuz** · March 30th 2011, 12:02 AM

Originally Posted by mathman88

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

It is.

We have to show that $\dfrac{f_{1}(x)+g_{1}(x)}{f_{2}(x)+g_{2}(x)}\sim1$ .
and we know that $\dfrac{f_{1}(x)}{f_{2}(x)}\sim1$ and $\dfrac{g_{1}(x)}{g_{2}(x)}\sim1$ .
Therefore, we have
$\dfrac{f_{1}(x)+g_{1}(x)}{f_{2}(x)+g_{2}(x)}=\dfra c{1+g_{1}(x)/f_{1}(x)}{\big(f_{2}(x)/f_{1}(x)\big)+\big(g_{2}(x)/f_{1}(x)\big)}$
........................ $\sim\dfrac{1+\big(g_{1}(x)/f_{1}(x)\big)}{1+\big(g_{2}(x)/f_{1}(x)\big)}$
........................ $=\dfrac{\big(f_{1}(x)/g_{1}(x)\big)+1}{\big(f_{1}(x)/g_{1}(x)\big)+\big(g_{2}(x)/g_{1}(x)\big)}$
........................ $\sim\dfrac{\big(f_{1}(x)/g_{1}(x)\big)+1}{\big(f_{1}(x)/g_{1}(x)\big)+1}\equiv1$ .

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