Consider the function $f(x) = \frac {ax + b} {cx + d}$ with $a,b,c,d \in \mathbb R, \neq 0$. If $ad \neq cb$, then as $x \to \frac {-d} {c}$, then $f(x)$ approaches $+ \infty$ from one side and $- \infty$ from the other side. (Which side is which depends on if $ad < cb$ or $ad > cb$.
What is the proper way to state $f(x)$ approaches $+ \infty$ from one side and $- \infty$ from the other side? Saying $x = \frac {-d} {c}$ is an asymptote doesn't convey that it approaches both positive and negative infinity.
And, what is the simplest way to prove it? Since $f$ is continuous everywhere but $\frac {-d} {c}$, and monotonic everywhere, then $f$ must be above a line $y = k$ on one side of $\frac {-d} {c}$, and jump to below that line on the other side ($k$ happens to be $\frac a c$). But that doesn't prove that $f$ approaches $\pm \infty$ on either side. I could work this out algebraically, but the case analysis (for positive or negative values of $a,b,c,d$) gets cumbersome, and my intuition is that there is a simple proof. And what is the correct way to
The graph of the function $$f(x)=\frac{ax+b}{cx+d}$$ is a hyperbola so it approaches to $\pm\infty$
To prove it
$$f(x)=\frac{ax+b}{cx+d}$$ $$=\frac{a(x+\frac{d}{c})+b-\frac{ad}{c}}{cx+d}$$ $$=\frac{a}{c}+\frac{b-\frac{ad}{c}}{cx+d}$$ $$=\frac{a}{c}+\frac{\frac{b}{c}-\frac{ad}{c^2}}{x+\frac{d}{c}}$$ I hope you know that is the equation of a hyperbola. The vertical asymptote is $x=\frac{-d}{c}$ and the horizontal asymptote is $\frac{a}{c}$