SAT1000 - P880

For function $f(x), g(x)$ which have the same domain $D$ , if there exists $h(x)=kx+b$ ( $k,b$ are constant), so that:

$\forall m \in (0,+\infty), \exists x_0 \in D, \forall x \in D \wedge x>x_0, 0<f(x)-h(x)<m, 0<h(x)-g(x)<m,$

Then line $l$ : $y=kx+b$ is called the bipartite asymptote for curve $y=f(x)$ and $y=g(x)$ .

Here are four groups of functions which are defined at $(1,+\infty)$ :

1. $f(x)=x^2, g(x)=\sqrt{x}$ .

2. $f(x)=10^{-x}+2, g(x)=\dfrac{2x-3}{x}$ .

3. $f(x)=\dfrac{x^2+1}{x}, g(x)=\dfrac{x \ln x+1}{\ln x}$ .

4. $f(x)=\dfrac{2x^2}{x+1}, g(x)=2(x-1-e^{-x})$ .

Which groups of curve $y=f(x)$ and $y=g(x)$ has a bipartite asymptote ?

How to submit:

Let $p_1, p_2, p_3, p_4$ be the boolean value of the group $1,2,3,4$ , if group $k$ has a bipartite asymptote , $p_k=1$ , else $p_k=0$ .

Then submit $\displaystyle \sum_{k=1}^4 p_k \cdot 2^{k-1}$ .

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Have a look at my problem set: SAT 1000 problems