real analysis - Finding bijection from (0,1) → N

How exactly do I go about finding a bijection between (0,1) → N \ {0}

so $(0,1) → (1, \infty)$. I figured I could look at this as finding a function from $(0,1) → (0, \infty)$ and just adding 1.

I've seen examples where f(x) = $\frac{1}{x} -1$ then $f(0) = \infty$ and $f(1) = 0$ (but these were on closed sets)

I couldn't find an example of a function such that $\lim_{x\to 1} = \infty$ or $\lim_{x\to 0} = \infty$ which is what it looks like I need here.

Can someone give me an example, or a way to find such a function?

Answer

Here are two bijections $f$ from $(0,1)$ to $(1,\infty)$:

1) Let $f(x)=\frac{1}{1-x}$;

2) Let $f(x)=1+\tan\left(\frac{\pi x}{2}\right)$.