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  1. #1
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    Asymptotic

    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) ?
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  2. #2
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    If f_1\lesssim f_2 and g_1 \lesssim g_2, is f_1+g_1 \lesssim f_2 + g_2?
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathman88 View Post
    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?
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  4. #4
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    Quote Originally Posted by Drexel28 View Post
    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
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  5. #5
    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by mathman88 View Post
    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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