Problem :
Let $f(x)$ be a positive differentiable function on $x>0$ such that :
Let $x\in[0,\varepsilon]$,$0<\varepsilon<1$ such that :
$$f(x)\simeq ax+b$$
And :
$$\lim_{x\to\infty}f(x)-cx-d=0$$
I can exhibit a such function see :
$$f\left(x\right)=\frac{x}{\left(x^{x}\ln\left(x^{x}+1\right)\right)^{\frac{1}{x^{x}}}}$$
But it's a very exotic function .
Question :
Can you provide some simple example for $f(x)$ ?
Basically we want $f(x)$ to behave like $ax+b$ for small $x$ and $cx+d$ for large $x$, therefore we can define a 'switch' function $s(x)$ such that $s(x)\approx1$ for small $x$ and $s(x)\approx0$ for large $x$, then let $$f(x)=s(x)(ax+b)+(1-s(x))(cx+d).$$ A simple example would be $s(x)=\frac1{1+x^2}$, which gives $$f(x)=\frac{ax+b}{1+x^2}+\left(1-\frac1{1+x^2}\right)(cx+d)=\frac{cx^3+dx^2+ax+b}{1+x^2}.$$ Here is a Desmos plot to visualise this function.