I will try to give you a general formula you can use for each of those.
The so called homographic functions, with $ad-cb \neq 0$
$y=\frac{ax+b}{cx+d}$
have the following general properties:
$f(0)=\frac{b}{d}$
$\lim\limits_{x \to -\frac d c} f(x) = \infty $ $\lim\limits_{x \to \infty} f(x) = \frac a c$
Maybe, a more natural form is the canonical one, where we have
$y=k+\frac{a}{x-h}$
The properties are now even more evident,
$\lim\limits_{x \to \infty} f(x) = k$
$\lim\limits_{x \to h} f(x) = \infty $
and $f(0)=k-\frac a h $
Their complex analog, the Möbius transformations, play an important role in complex analysis.
Also, given any continuous function $g(x)$, we can create a removable discontinuity by defining
$G(x) = \begin{cases} g(x) \text{ ; } x \neq a \cr M \text{ ; } x =a \end{cases}$
where $M \neq g(a)$.