How do we know that a rational function has at most 1 slant asymptote? I was wondering why it must be the case that a rational function such as
$$f(x) = 3x + 2 + \frac{2}{x+6}$$
has only one slant asymptote. My textbook gave an intuitive explanation of why this might be true but it was far from rigorous. I understand why
$$y = 3x + 2$$
is a slant asymptote. But isn't it possible that it could have more? Could anybody explain why it must be the case that functions such as $f$ must have at most 1 slant asymptote?
 A: There are 3 types of asymptotes:

*

*Vertical

*Horizontal

*Slant
So, non vertical asymptote can be slant asymptote or horizontal asymptote.
A function $f(x)$ can have at most two non vertical asymptotes:

*

*First one is for the case when $x\to-\infty$.

*Second one is for the case when $x\to+\infty$.

When a function has only one non vertical asymptote? It can happen only in these 3 cases:

*

*Function $f(x)$ has asymptote for the case $x\to-\infty$, but doesn't have asymptote for the case $x\to+\infty$.

*Function $f(x)$ doesn't have asymptote for the case $x\to-\infty$, but has asymptote for the case $x\to+\infty$.

*Function $f(x)$ has the same asymptote in both cases: $x\to-\infty$ and $x\to+\infty$.

Function $f(x)=3x+2+\frac{2}{x+6}$ has asymptote $y=3x+2$ when $x\to-\infty$ since the term $\frac{2}{x+6}$ becomes very small when $x$ is very large negative number (so $f(x)=3x+2+\frac{2}{x+6}\approx 3x+2$, when $x\to-\infty$) and the degree of $3x+2$ is one, so it's a line and thus it's an asymptote (if this part would be a polynomial of higher degree than $1$ then asymptote wouldn't exist).
But it also has the same asymptote $y=3x+2$ when $x\to+\infty$ (since $\frac{2}{x+6}$ becomes also very small when $x$ is very large positive number). That's why this function has only one non vertical asymptote, which is slant asymptote in this case (it would be a horizontal asymptote if degree of the part mentioned above would be zero, i.e. it would be a constant).
Bonus tip: a function can have unlimited number of vertical asymptotes (for example, $f(x)=tg(x)$ has vertical asymptotes at points $x=\frac{\pi}{2}+k\pi, k\in\mathbb{Z}$)
