应用极限求函数的对称中心

$求函数f(x)=\cfrac{2}{e^x+1},\quad g(x)=\cfrac{3e^x+1}{e^x+1}的对称中心(a,b)$
$由2b=f(+\infty)+f(-\infty)求出b;$
$再由f(a)=b，求出a即可，此法成立条件是x\in \mathbb{R}$
$\lim_{x \to +\infty} \cfrac{3e^x+1}{e^x+1} =3$
$\lim_{x \to -\infty} \cfrac{3e^x+1}{e^x+1} =\cfrac{1}{1}=1$
$2b=4\Rightarrow b=2$
$f(a)=b=2\Rightarrow \cfrac{3e^x+1}{e^x+1} =2\Rightarrow x=0$
${\color{Violet} \therefore 对称中心为(0,2)}$

$g(x)=\cfrac{2}{e^x+1}\Rightarrow g(+\infty) =0,g(-\infty) =2,{\color{Red} b=1}$
$f(a)=1=\cfrac{2}{e^x+1} {\color{Red} \Rightarrow a=0}$