Finding function with specified limits

Question

I am seeking a function $f(x)$ such that $f(x)\to 1$ as $x\to 0$, and $f(x)\to \alpha>0$ as $x\to \infty$. I thought the function must have a term proportional to $e^{-x}$, but I can't see what the entire function may look like. After repeated failed attempts, I am beginning to wonder if such a function exists. Any help is appreciated. Thanks.

Answer 1

Hint

$$f(x)=\frac{1}{x^2+1}$$ goes to $1$ at $x=0$ and to $0$ at infinity.

Look for $$g(x)=af(x)+b$$

Answer 2

The function $$f(x)=\frac{\alpha x+1}{x+1}$$ satisfies $f(0)=1$ and $$\lim_{x\rightarrow \infty} f(x)=\alpha$$